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All Textbook Solutions for Discrete Mathematics With Applications

A universal statement asserts that a certain property is _______for ________A conditional statement asserts that if one thing____ then some other thing___Given a property that may or may not be true, an existensial statement asserts that ________for which the property is true.In each of 1—6, fill in the blanks using a variable or variables to rewrite the given statement. Is there a real number whose square is -1? a. Is there a real number x such that _____ ? b. Does there exist _____ such that x2=1 ?In each of 1—6, fill in the blanks using a variable or variables to rewrite the given statement. Is there an integer that has a remainder of 2 when it is divided by 5 and a remainder of 3 when it is divided by 6? a. Is there an integer n such that n has ? b. Does there exist _____ such that if n is divided by 5 the remainder is 2 and if _____? Note. There are integers with this property, can you think of one?In each of 1—6, fill in the blanks using a variable or variables to rewrite the given statement. Given any two distinct real numbers, there is a real number in between them. a. Given any two distinct real numbers a and b, there is a real number c such that c is _____ b. For any two _____ _____ such that c is between a and b.Given any real number, there is a number that is greater. a. Given any real number r, there is ______s such that s is______ b. For any _____, ________such that x > r The reciprocal of any postive real number is positive. Given any positive real number is positive. Given any positive real number r, the reciprocal of_____, For any real number r, if r is_________, then_____ If a real number r________ then _______6ES Rewrite the following statements less formally, Without using variables. Determine, as best as you can, whether the statements are true or fales. There are real numbers u and v with the property that u+vuv . There is a real number x such that x2x . For every positive integer n,n2n . For all real numbers a and b,a+ba+b For every object J, if J is a square then J has four All squares_____ Every square______ If an object is a square, then it_______ If J_______, then J______ For every sauare J,________For every equation E, if E is quadratic then E has at most two real solutions. All quadeatic equation____. Every quadratic equation_____ If an equation is quadraticm then it______ If E______, then E________ For every quadratic equation E,_______Every nonzero real number has a reciropal. All nonzero real numbers______, For every nonzeros real number r, theer is ______for r, For every nonzero real number r, there is a real number s such that______Evaery positive number has a positive square root. All positive number_______. For every positive number e, there is ______for e. For every postive number e, there is a positive number r such that ______There is a real number whose product with every number leaves the number unchanged. Some __has the property that its________. There is a real number r such that the product of r_______ There is a real number r with the property that for real number s,___There is a real number whose product with ever real number equals zero. Some ________has the property that its________. There is a real number a such that the product of a___ There is a real number a with the property that for every real number b,______When the elements of a set are given using the set-roster notation, the order in which they are listed___.The symbol R denotes ____.The symbol Z denotes ______The symbol Q denotes__The notation {xP(x)} is read _______6TY 7TY Given sets A,B, and C, the Cartesian production ABC is ________A string of length n over a set S is an ordered n-tuple of elements of S, written without ________or _______1ES Write in words how to read each of the following out loud. {xR+0x1} {xRx0orx1} {nznisafactorof6} {nZ+nisafactorof6}Is 4={4}? How many elements are in the set {3,4,3,5} ? How many elements are in the set {1,{1}, {1{1}}}?a. Is 2{2}? b. How many elements are in the set {2,2,2,2} ? c. How many elements are in the set {0,{0}} ? d. Is {0}{{0},{1}}? e. Is 0{{0},{1}} ?Which of the following sets are equal? A={0,1,2}B={xR1x3}C={xR1x3}D={xZ1x3}E={xZ1x3}For each integer n, let Tn={n,n2} . How many elements are in each of T2,T3,T1 , and T0 ?Justify your answer.7ES 8ES Is3{1,2,3}? Is 1{1}? Is {2}{1,2}? Is {3}{1,{2},{3}}? Is 1{1}? Is {2}{1,{2},{3}}? Is {1}{1,2} ? Is 1{{1},2} ? Is {1}{1,{2}}? Is {1}{1}?Is ((2)2,22)=(22,( 2)2)? Is (5,5)=(5,5)? Is (89,13)=(1,1)? Is (24( 2)3)=(368)?11ES 12ES 13ES 14ES Let S={0,1} . List all the string of length 4 over S that contain three or more 0’s.Let T={x,y} . List all the strings of length 5 over T that have exactly one y.Given sets A and B , relation from A to B is ____A function F from B is a relation from A to B that satisfies the following two properties: a. for every element x of A, there is________ b. for all elements x in A and y and z in B, if ________then______.If F is a function from A to B and x is an element of A , then F(x) is_____Let A={2,3,4} and B={6,8,10} and define a relation R from A to B as follows: For every (x,y)AB , (x,y)R means that yx is an integer. Is 4 R 6? Is 4 R 8? Is (3,8)R? Is (2,10)R? Write R as a set of ordered pairs. Write the domain and co-domain of R. Draw an arrow diagram for R.Let C=D={3,2,1,1,2,3} and define a elation S from C to D as follows: For every (x,y)CD . (x,y)S means that 1x1y is an integer. Is 2 S 1? Is -1 S -1? Is (3,3)S ? Is (3,3)S Write S as a set of ordered pairs. Write the domain and co-domain of S. Draw an arrow diagram for S.Let E={1,2,3} and F={2,1,0} and define a relation Tfrom E to F as follows: For every (x,y)EF. (x,y)Tmeansthatxy3isaninteger. Is 3T 0? Is 1T(1) ? Is (2,1)T ? Is(3,2)T? Write T as a set of ordered pairs. Write the domain and co-domain of T. Draw an arrow diagram for T.Let G=-2,0,2) and H=4,6,8) and define a relation V from G to H as follows: For every (x,y)GH, (x,y)V means that xy4 is an integer. a. Is 2 V 6? Is (-2) V (8)? Is (0,6)V? Is (2,4)V? b. Write V as a set of ordered pairs. c. Write the domain and co-domain of V. d. Draw an arrow diagram for V.Define a relations S from R to R as follows: For every (x,y)RR , (x,y)Smeansthatxy. Is (2,1)S? Is (2,2)S? Is 2S3? Draw the graph of S in the Cartesian plane.Define a relation R from R to R as follows: For every (x,y)RR, (x,y)Rmeansthaty=x2 Is (2,4)R? Is (4,2)R?Is (-3) R 9? Is 9 R (-3)? Draw the grah of R in the Cartesian plane.Let A={4,5,6} and B={5,6,7} and define relations R,S, and T from A to B as follows: For every (x,y)AB: (x,y)R means that xy . (x,y)Smeansthatxy2isaninteger. T={(4,7),(6,5),(6,7)}. Draw arrow diagrams for R, S, and T. Indicate whether any of the relations R, S, and T are functions.Let A={2,4} and B={1,3,5} and define relations U, V, and W from A to B as follows: For every (x,y)AB: (x,y)U means that yx2 . (x,y)V means that y1=x2 . w={(2,5),(4,1),(2,3)} . a. Draw arrow diagrams for U, V, and W. b. Indicate whether any of the relations U, V, and Ware functions.Find all function from {01,} to {1} . Find two relations from {0,1} to {1}that are not functions.Find tour relations from {a,b} to {x,y} that are not function from {a,b} to {x,y} .Let A={0,1,2} and let S be the set of all strings over A. Define a relation L from S to Znonnesx as follows: For every string s in S and every nonnegative integer n. (s,n)L means that the length of s is n. Then L is a function because every string in S has one and only one length. Find L(0201) and L(12).Let A={x,y} and let S be the set all strings over A. Define a relation C from S to S as follows: For all strings s and t in S, (s,t)Cmeansthatt=ys Then C is a function because every string in S consists entirely of x’s and y’s and adding an additional y on the left creates a single new string that consists of x’s and y’s and is, therefore, also in S. Find C(x) and C(yyxyx).Let A={1,0,1} and B={t,u,v,w} . Define a function F:AB by the following arrow diagram: Write the domain and co-domain of F. Find F(1),F(0), and F(1) .Let C = (1,2,3,4) and D={a,b,c,d}. Define a function G:CD by the following arrow diagram: Write the domain and co-domain of G. Find G (1), G(2), G(3), and G(4).Let X=2,4,5) and Y=(1,2,4,6) . Which of the following arrow diagrams determine functions from X to Y’?Let f be the squaring function defined in Example 1.3.6. Find f(1),f(0) , and f(12) .Let g be the successor function defined in Example 1.3.6. Find g(1000),g(0), and g(999) .Let h be the constant function defined in Example 1.3.3. Find h(125),h(01) , and h(917) .Define functions f and g from R to R by the following formulas: For every xR , f(x)=2xandg(x)=2x3+2xx2+1 Does f=g ? Explain.Define functions H and K from R to R by the following formulas: For every xR . H(x)=(x2)2 and K(x)=(x1)(x3)+1. . Does H = K? Explain.A graph consists of two finite sets: ______and ______, where each edge is associated with a set consisting of ______.A loop in a graph is_____Two distinct edges in a graph are parallel if, and only if,________Two vertices are called adjacent if, and only if, _______.An edge is incident on _______Two edges incident on the same endpoint are_________A vertex on which no edges are incident is________8TY 9TY In 1 and 2, graphs are represented by drawings Define each graph formally by specifying its vertex set, its edge set, and a table giving the edge-endpoint function.In 1 and 2, graphs are represented by drawings. Define each graph formally by specifying its vertex set. Its edge set, and a table giving the edge-endpoint functions.In 3 and 4, draw pictures of the specified graphs. Graph G has vertex set {v1,v2,v3,v4,v5} and edge set {e1,e2,e3,e4} , with edge-endpoint function as follows:4ES 5ES In 5-7, show that the two drawings represent the same graaph by labeling the vertices and edges of the right-hand drawing to correspond to those of the left-hand drawing.In 5-7, show that the two drawings represent the same graaph by labeling the vertices and edges of the right-hand drawing to correspond to those of the left-hand drawing.For each of the graphs in 8 and 9: (i) Find all edges that are incidents to v1 . (ii) Find all vertices that are adjacent to v3 . (iii) Find all edges that are adjacent to e1. (iv) Find all loops. (v) Find all parallel edges. (vi) Find all isolated vertices. (vii) Find the degree of v3 ..For each of the graphs in 8 and 9: (i) Find all edges that are incidents to v1 . (ii) Find all vertices that are adjacent to v3 . (iii) Find all edges that are adjacent to e1. (iv) Find all loops. (v) Find all parallel edges. (vi) Find all isolated vertices. (vii) Find the degree of v3 .Use the graph of Example 1.4.6 to determine Whether Sports illustrated contains printed writing: Where Poetry Magazine contains long words.Find three other winning sequences of moves for the vegetarians and the cannibals in Example 1.4.7.Another famous puzzle used as an example in the study of artificial intelligence seems first to have Solve the vegetarians-and-cannibals puzzle for the case where there are three vegetarians and three cannibals to be transported from one side of a river to the other.Two jugs A and B have capacities of 3 quarts and 5 quarts respectively. Can you use the jugs to measure out exactly 1 quart of water, while obeying the following restrictions? You may fill either jug to capacity restrictions? You may fill either jug to capacity from a water tap; you may empty the contents of either jug into a drain; and you may pour water from either jug into the other.15ES In this exercise a graph is used to help solve a scheduling problem. Twelve faculty members in, a mathematics department serve on the following committees: Undergraduate Education: Tenner. Peterson. Kashina, Degras Graduate Education: Hu. Ramsey Degras. Bergen Colloquium. Carroll. Drupaieski. Au-Yeung Library: Ugarcovici. Tenner. Carroll Hiring: Hu. Drupieski. Ramsey. Peterson Personnel: Ramsey, Wang Ugarcovici The committees must all meet during the first week of classes, but there are only three time slots available. Find a schedule dial will allow all faculty members to attend I he meetings o! all committees on which they serve. To do this, represent each committee as the vertex of a graph, and draw an edge between two vertices if the two committees have a common member Find a way to color the vertices using only three colors so that no two committees have the same color, and explain how to u*e the result to schedule the meetings..A deptnn1 war to ithechik final ezans that no %aude1 has nre than o. ezam on an gsn day. I1 wrtices at the grd* bek s& the -uures that are being taken by nxe than ( SLU&SL wh an edge connecting to wrikes if there as a tu&nt in both coure. Find a ay to cdor she wrtces at the graph wLih only four coI ars 54) thai no i Ijaces wrtes haw the san cdui and eispban bo to ue the rewlt to s.-hekile the f anal ezams.An and statement is true when, and only when, both components are______An or statement is false when, and only when, both components are________.Two statement forms are logically equivalent when, and only when, they always have ______ .,De Morgan’s laws say (1) that the negation of an and statement is logically equivalent to the _______ statement in which each component is _______ and (2) that the negation of an or statement is logically equivalent to the _______ statement in which each component is _______A tautology is a statement that is always _____.A contradiction is a statement that is always _____In eachof 1—4 represent the common form of each argument using letters t stand for component sentences, and fillin the blank so that the argument in part (b) has the samelogical form as the argument in part (a). a. If all Integers are rational, then the number is rational. All integers are rational. Therefore, the number1 is rational. b. If all algebraicexpressions can be written in prefix notation, then____________ Therefore. (a+2b)(a2b) can be written in prefix notation.In each of 1-4 represent the common form of each argument using letters to stand for component sentences, and fill in the blanks so that the argument in part (b) has the same logical from as the argurnent in part (a) If all computer programs contain errors, then this program contains an error. This program contains an error. Therefore, if is not the case that all computer Programs contain errors. If___then___2 is not odd. Therefore, it is not the case that all prime numbers are odd.In each of 1—4 represent the common form of each argument using letters to stand for component sentences, and fill in the blanks so that the argument in part (b) has the same logical form as the argument in part (a) This number is even or this number is odd. This number is not even. ____or logic is confusing . My mind is not shot. Therefore, _____In each of 1—4 represent the common form of each argument using letters to stand for component sentences, and fill m the blanks so that the argument in part (b) has the same logical form as the argument in part (a) 4. a. If the program sy ntax as faulty, then the computer will generate an error message then the program will not run. Thrrefore. if the program syntax is faulty, then the program w ill not run. b. If this simple graph then it is complete 11 tins graph then any two of its vertices can be joined by a path. Therefore, if this simple graph has 4 vertices and 6 edges. then Indicate which of the following sentences are statements. a. 1,024 is the smallest four-digit number that is a perfect square. b. She is a mathematics major. c. 128=26 d. x=26 Write the statements in 6-9 in symbolic form using the symbols ~Vand and the indicated letter to represent component statements. Let s= “stocks are increasing” and i=”interest rates are steady. Stocks are increasing but interest rates are steady. Neither are stocks increasing nor are interest rates steady.Write the statements in 6-9 in symbolic form using the symbols ~,V and A and the indicated letters to represent component statements. Juan is a math major but not a computer science major. (m=”Juan is a math major,” c=”Juan is a computer science major”)Write the statements in 6-9 n symbolic form using the symbols ~,V and and the indicated let ted to represent component staternents. Let h=”John is heakthy,” w= “John is wealthy,” and s= “John is wise.” John is healthy and wealthy but not wise. John is not wealthy but he is healthy and wise. John is neither wealthy nor wise, but he is healthy.Write the statements in 6-9 in symbolic form using the symbols ~V, and A and the indicated to represent component statements. Let p=x5,q=x=5 and r=10x. x5 10x5 10x5 Let p be the statement "DATAENDFLAG is off," q the statement “ERROR equals 0." and r the statement "Sum is less than 1,000." Express the following sentences in symbolic notation. DATAENDFLAG is off, ERROR equal 0, and SUM is less than 1,000. DATAENDFLAG is off but ERROR is not equal to 0. DATAENDFLAG is off; however, ERROR is not 0 or SUM is greater than or equal to 1,000. DATAENDFLAG is on and ERROR equals 0 but SUM is greater than or equal to 1,000. Either DATAENDFLAG is on or it is the case that both ERROR equals 0 and SUM is less than 1,000.In the following sentence, is the word or used in its inclusive or exclusive sense? A team wins the playoffs if it wins two games in a row or a total of three games.Write truth tables for the statement forms in 12-15. pq Write truth tables for the statement forms in 12-15. ~(pq)(pq)Write truth tables for the statement forms in 12-15. p(qr)Write truth tables for the statement forms in 12-15. p(qVr)Determine whether the statement forms in 16—24 are logically equivalent. In each case, construct a truth table arid include a sentence justifying your answer. Your sentence should show that you understand the meaning of logical equivalence. p(pq) and p Determine whether the statement forms in 16-24 are logically equivalent. In each construct a truth table and include a sentence justifying your answer. Your sentence should show that you understand the meaning of logical equivalence. ~(pq)andp~q Determine whether the statement forms in 16—24 are logically equivalent. In each case, construct a truth table arid include a sentence justifying your answer. Your sentence should show that you understand the meaning logical equ1vence.. ptandt Determine whether the statement forms in 16—24 are logically equivalent. In each case, construct a truth table arid include a sentence justifying your answer. Your sentence should show that you understand the meaning logical equ1vence.. pt and p Determine whether the statement forms in 16—24 are logically equivalent. In each case, construct a truth table arid include a sentence justifying your answer. Your sentence should show that you understand the meaning logical equ1vence.. pcandpe Determine whether the statement forms in 16-24 are logically equivalent. In each case, construct a truth table and include a sentence justifying your answer. Your sentence should show that you understand the meaning of logical equivalence. (pq)randp(qr)Determine whether the statement forms in 16-24 are logically equivalent. In each case, construct a truth table and include a sentence justifying your answer. Your sentence should show that you understand the meaning of logical equivalence. p(qr)and(qq)(pr)Determine whether the statement forms in 16-24 are logically equivalent. In each case, construct a truth table and include a sentence justifying your answer. Your sentence should show that you understand the meaning of logical equivalence. (pq)randp(qr)Determine whether the statement forms in 16-24 are logically equivalent. In each case, construct a truth table and include a sentence justifying your answer. Your sentence should show that you understand the meaning of logical equivalence. (pq)(pr)and(pq)r Use De Morgan’s laws to write negations for the statements in 25-30. Hal is math major and Hal’s sister is a computer science major.Use De Morgan’s laws to write negations for the statements in 25-30. Sam is an orange belt and Kate is a red belt.Use De Morgan’s laws to write negations for the statements in 25-30. The connector is loose or the machine is unplugged.Use De Morgan’s laws to write negations for the statements in 25-30. The train is late or my or watch is fast.Use De Morgan’s laws to write negations for the statement in 25-30. This copmputer program has a logical error in the first ten lines or it is being run with an incomplente data set.Use De Morgan’s laws to write negations for the statements in 25-30. The dollar is at an all-time high and the stock market is at a record low.31ES Assume x is a particular real number and use De Morgan’s laws to write negations for the statements is 32-37. 2x7 Assume x is a particular real number and use De Morgan’s laws to write negations for the statements is 32-37. 10x2 Assume x is a particular real number and use De Morgan’s laws to write negations for the statements in 32-37. x2orx5 Assume x is a particular real number and use De Morgan’s laws to write negations for the statements in 32-37. x1orx1 Assume x is a particular real number and use De Morgan’s laws to write negations for the statements is 32-37. 1x3 Assume x is a particular real number and use De Morgan’s laws to write negations for the statements is 32-37. 0x7 In 38 and 39, imagine that num_orders and num_instock are particular values, such as might occur during execution of a computer program. Write negations for the following statements. (numorder100andnuminstock500)ornuminstock200 In 38 and 39, imagine that num_orders and num_instock are particular values, such as might occur diving execution of a cornputer program. Write negations for the following statements. (numorders50andnuminstock300)or(50numorders75andnuminstock500)Use truth to establish which of the statement forms in 40-43 are tautologies and which are contradictions. (pq)(~p(p~q))Use truth tables to establish which of the statement forms in 40-43 are tautologies and which are contradictions. (p~q)(~qq)Use truth to establish which of the statement forms in 40-43 are tautologies and which are contradictions. ((~pq)(qr))~q Use truth tables to establish which of the statement forms in 40-43 are tautologies and which are contradictions. (~pq)(p~q)Recall that axb means that ax and xb . Also ab means that ab or a=b . Find all real numbers that satisfy the following inequalities. 2x0 1x1 Determine whether the statements in (a) and (b) are logically equivalent. Bob is both a math and computer science major and Ann is a math major, but Ann is not both a math and computer science major. It is not the case that both Bob and Ann are both math and computer science majors, but it is the case that Ann is a math major and Bob is both a math and computer science major.Let the symbol denote exclusive or; so pq=(pVq)(pq) . Hence the truth table for pqis as follows: Find simpler statement forms that are logically equivalent to pqand(pp)p. Is (pq)r=p(qr) ? Justify your answer Is (pq)r=(pr)(qr) ? Justify your answer.In logic and in standard English, a double negative is equivalent to a positive. There is one fairly common English usage in which a”double positive” is equivalent to a negative. What is it? Can you think of others?In 48 and 49 below, a logical equivalence is derived from Theorem 2.1.1. Supply a reason for each step. (p~q)(pq)=p(Qqq)by(a)=p(q~q)by(b)=ptby(c)=pby(d) Therefore, (p~q)(pq)=p.In 48 and 49 below, a logical equivalence is derived from Theorem 211. Supply a reason for cacti step. (p~q)(p~q)=(~qp)(~qv~p)by(a)=qeby(b)=qby(c)Use Theorem 2.11 to verify the logical equivalences in 50-54. Supply a reason for each step. (p~q)p=p Use theorem 2.11 to verify the logical equivalences in 50-54, Supply a reason for each step. p(~qp)=p Use Theorem 2.11 to verify the logical equivalences in 50-54. Supply a reason for each step. ~(p~q)(~p~q)=~p Use Theorem 2.11 to verify the logical equivalences in 50-54. Supply a reason for each step. ~((~pq)v(~p~q))(qq)=p Use Theorem 2.11 to verify the logical equivalences in 50-54. Supply a reason for each step. (p(~( ~pq)))(pq)=p An if-then statement is false if, and only if, the hypothesis is _______and the conclusion is___The negation of “if p then q” is _____The converse of”if p then q” is _______The contrapositive of “if p the q” is _________5TY A conditional statement and its contrapositive are_______7TY “R is a sufficient condition for S” means “if ______then_________”“R is a necessary condition for S” means “if _______then__________”10TY Rewrite the statements in 1-4 in if-then form.Rewrite the statements in 1-4 in if-then from. I am on time for work if I catch the 8:05 bus.Rewrite the statements in 1-4 in if-then form. Freeze or I’ll shoot.4ES Construct truth tables for the statements forms in 5-11. pVqq Construct truth tables for the statements forms in 5-11. (pq)(~pq)q 7ES 8ES Construct truth tables for the statements forms in 5-11. prqvr 10ES 11ES Use the logical equivalence established in Example 2.2.3, p V q —, r (p —, r) A (q —, r), to rewrite the following statement. (Assume that x represents a fixed real number.) If x > 2 or x< —2, then x 2>4.13ES Show that the following statement forms are all logically equivalent: pqr,p~qr,andp~rq b. Use the logical equivalences established in part (a) to rewrite the following sentence in two different ways. (Assume that n represents a fixed integer.) If n is prime, then n is odd or n is 2.Determine whether the following statement forms are logically equivalent: P(qr)and(pq)r 16ES In 16 and 17, write each o the two statements in symbolic form arid determine whether they are logically equivalent. Include a truth and a few words of explanation to show that you understand what it means for statements to be logically equivalent. If 2 is a factor of n and 3 is a factor of n, then 6 is a factor of n. 2 is not a factor of 3 is not a factor of n or 6 is a factor of n.Write each at the following three statements in symbolic from and determine which parirs are logically equivalent. Include truth tables and a few words of explanation. If it walks like a duck and it talks like a duck, then it is duck, Either it does not walk like a duck or it does not talk like a duck, or it is a duck. If it does not walk like a duck and it does not talk like a duck, then it is not a duck.True or false? The negation of “If Sue is Luiz’s mother, then Ali” is his cousin” is “If Sue is Luiz’s mother, then Ali is his cousin” is “If Sue is Luiz’s mother, then Ali is not his cousin.”Write negations for each of the following statement. (Assume that all variables represent fixed quantities or entities, as approproiate.) If P is a square, then P is a rectangle. If today is New Year’s Eve, then tomorrow is January. If the decimal expansion of r is terminationg, then r is rational. If n is prime, then n is odd or n is 2. If x is nonnegative, then x is positive or x is 0. And Sue is her aunt. If Tom is Ann’s father, then Jim is her uncle and Sue is her aunt. If n is divisible by 6, then n is divisible by 2 and n divisible by 3.Suppose that p and q are statements so that p ) q is false. Find the truth values of each of’ the following: ~pq pq qp Write negations for each of the following statements. (Assume that all variables represent fixed quantities or entities, as appropriate.) If P is a square, then P is a rectangle If today is New Year’s Eve, then tomorrow is January. If the decimal expansion of r is terminating, then r is rational. If n is prime, then n is odd or n is 2. If x is nonnegative, then x is positive or x is 0. If Toni is Ann’s father, then Jim is her untie and Sue is her aunt. If n is divisible by 6, then n is divisible by 2 and n is divisible by 3 Write negations for each of the following statements. (Assume that all variables represent fixed quantities or entities, as appropriate.) If P is a square, then P is a rectangle If today is New Year’s Eve, then tomorrow is January. If the decimal expansion of r is terminating, then r is rational. If n is prime, then n is odd or n is 2. If x is nonnegative, then x is positive or x is 0. If Toni is Ann’s father, then Jim is her untie and Sue is her aunt. If n is divisible by 6, then n is divisible by 2 and n is divisible by 3. 23. Write the converse and inverse for each statement of exercise 20.24ES 25ES Use truth tables to establish the truth of each statement in 24-27. A conditional statement and its contrapositive are logically equivalent to each other.27ES 28ES If statement forms P and Q are logically equivalent, then PQ is a tautology. Conversely if PQ is a tautology, then P and Q are logically equivalent, Use to convert each of the logical equivalences 29-31 to a tautogy. Then use a true table to verify each tautology. p(qr)=(p~q)r 30ES If statement forms P mid Q are logically equivalent, then PQ is a tautology. Conversely, if PQ is a tautology, then P and Q ate logically equivalent. Use to convert each of the logical equivalences in 29—31 to a tautology. Then use a truth table to verify each tautology. p(qr)=(pq)r Rewrite each of the statements in 32 and 33 as a conjunct ion of two if-then statements. 32. This quadratic equation has two distinct real roots if, and only if, its discriminant is greater than zero.33ES Rewrite the statements in 34 and 35 in if-then form in two ways, one of which is the contrapositive of the other. Use the for formal definition of “only if.” The Cubs will win the pennant only if they win tomorrow’s game.Rewrite the statements in 34 and 35 en in-then form in two ways, one of which is the contrapositive of the other. Use the formal definition of “only” if. Samwill be allowed on Signe’s racing boat only if he is an expert sailor.Taking the long view on u education, you go to the Prestige Corporation and ask what you should do in college and ask what you should do in college to be hired when you graduate. The personnel director personnel director replices that you will be hired only if you major in mathematics or computer science, get a B average or better, and take accounting, You do, in fact, becomes a math major, get a B+ average, and take accounting. You return to Prestige Corporation, make a formal application, and are turned down. Did the the personnel dictor lie to you?Some prograrnming languages use statements of the form “r unless s” to mean that as long as s does not happen, then r will happen. More formally. In 37-39, rewrite the statement in if-then form. Payment will be made on fifth unless a new hearing is granted.Some programming languages use statements of the form r unless s to mean that as long as s does not happen, then will happen. More formally: In 37-39, rewrite the statements in if-then font. Ann will go unless it rains.39ES 40ES 41ES 42ES Use the contrapositive to rewrite the statements in 42 and 43 in if-then in two ways. Doing homeswork regularly is a necessary condition for Jim to pass the course.44ES Note that a sufficient condition lot s is r” means, r is a sufficient condition for and that a necessary condition for s is necessary is a necessary condition for . Rewrrte the statements in 44 and 45 in if- then for is. A necessary condition for this computer pogram be to be is that at not produuihae erri meag during Lr-aui.“If compound X is boiling, then its temperature must be at least 150C ." Assuming dial this statement is true, which of the following must also be true? If the temperature of compound X is at least 150C , then compound X is boiling. If the temperature of compound X is less than 150C , then compound X is not boiling. Compound X will boil only if its temperature is at least 150C . It compound X is not boiling, then its temperature is less than 150C . A necessary condition for compound X to boil is that its temperature be at least 150C . A sufficient condition for compound X to boil is that its temperature be at least 150C .In 47— 50(a)use the logical equivalences pq=~pq and pq=(~qp)(~qp) to rewrite the given statement forms without using the symbol or and (b) use the logical equivalence pq=~(~p~q) to rewrite each statement form using only and — p~qr In 47— 50(a)use the logical equivalences pq=~pq and pq=(~qp)(~qp) to rewrite the given statement forms without using the symbol or and (b) use the logical equivalence pq=~(~p~q) to rewrite each statement form using only and — p~qrVq In 47-50 (a) use the logical equivalences pq=~pq and andPq=(~pq)(~qp) to rewrite the given statement for ms without using the symbol or , statement for ms without using the symbol or , and (b) use the logical equivalence or , and (b) use the logical equivalence pq=~(~p~q) to rewrite each statement form using only A and ~.. (pr)(qr)In 47-50(a) use the logical equivalences pq=~pq and pq=(~pq)(~qp) to rewrite the given statement forms without using the symbol or and (b) use the logical equivalence pq=~(~pq) to rewrite each statement form using only and ~. (p(qr))((pq)r)Given any statement form, is it possible to find a logically equivalent form that uses only and A’? Justify your answer.For an argument to be valid means that every argument of the same from whose premises ___has a ___conclusion.For an argument to be invalid means that there is an argument of the same from whose premises____ and whose conclusion_____.3TY Use modus ponens at modus tollens to fill in the blanks in the arguments of 1-5 so as to produce valid inferences. If 2 is rational, then 2=a/b for some integers a and b. It is not true that 2=a/b for some integers a and b.Use modus ponens or modus tollens to fill in the blanks in the arguments of 1—5 so as to produce valid inferences. 2. If 1 — 0.99999. . ... is less than every positive real number, then it equals zero. .. The number 1 — 0.99999. . . equals zero.Use modus ponens or modus tollens to fill in the blanks in the arguments of 1-5 so as to produce valid inferences. If logie is easy, then I am a monkey’s uncle. I am not a monkey’s uncle.Use modus ponens at modus tollens to fill in the blanks in the arguments of 1-5 so as to produce valid inferences. 4. If this graph can be colored with three colors, then a can ‘colored with four colors. This graph cannot be colored with four colors.Use modus ponens or modus tollens to fill in the blanks in the arguments of 1—5 so as to produce valid inferences. If they were unsure of the address, then they would have telephoned. .. They were sure of the address.Use truth tables to determine whether the argument forms in 6-11are valid. Indicate valid. Indicate which columns represent the premises and which represent the conclusion, and include a sentence explaining how the truth table supports your answer. Your explanation should show that you understand what it means for a form of from of argument to be valid or invalid. pqqrpq 7ES Use truth tables to determine whether the argument forms in 6-11are valid. Indicate valid. Indicate which columns represent the premises and which represent the conclusion, and include a sentence explaining how the truth table supports your answer. Your explanation should show that you understand what it means for a form of from of argument to be valid or invalid. pqp~qqrr Use truth tables to determine whether the argument forms in 6-11are valid. Indicate valid. Indicate which columns represent the premises and which represent the conclusion, and include a sentence explaining how the truth table supports your answer. Your explanation should show that you understand what it means for a form of from of argument to be valid or invalid. pq~rp~q~qp~r Use truth tables to determine whether the argument forms in 6-11are valid. Indicate valid. Indicate which columns represent the premises and which represent the conclusion, and include a sentence explaining how the truth table supports your answer. Your explanation should show that you understand what it means for a form of from of argument to be valid or invalid. pqrr~rp~q Use truth tables to determine whether the argument forms in 6-11are valid. Indicate valid. Indicate which columns represent the premises and which represent the conclusion, and include a sentence explaining how the truth table supports your answer. Your explanation should show that you understand what it means for a form of from of argument to be valid or invalid. pqrp~rq~r Use truth table to show that the following forms of argument are invalid. pqqp(converseerror) pq~p~q(inverseerror)Use truth tables to show that the argument forms referred to in 13-21 are valid. Indicate which columns represent the premises and which represent the conclusion, and include a sentence explaining how the truth table supports your answer. Your explanation should show that you understand what it means fro a form for a from of argument to be valid. Modus tollens: pq~q~p 14ES 15ES 16ES 17ES Use truth table to show that the argument forms referred to in 13-21 are valid. Indicate which columns represent the premises and which represent the conclusion, and include a sentence explanining how the truth table supports your answer. Your explanation should show that you understand what it means for a form of argument to be valid. Example 2.3.5(a)19ES 20ES 21ES 22ES Use symbols to write the logical form of each argument in 22 and 23, and then use a truth table to test the argument for validity. Indicate which columns represent the premises and Which represent the conclusion, and include a few words of expalnation s1howing that you understand the meaning of validity. Oleg is a math major or Oleg is an economices major. If Oleg is a math major, then Oleg is required to take Math 362. Oleg is an economics major or Oleg is not required to take Math 362.Some of the argurnents in 24-32 are valid, whereas others exhibit the convene or the inverse error. Use symbols to write the logical form of each argument. If the argurnent is valid, identify the rule of inference that guarantees its validity. Otherwise, state whether the converse or the inverse error is made. If Jules solved this problem correctly, then Jules obtained the answer 2 Jules solved this problem correctly.25ES Some at the arguments in 24—32 are valid, whereas others exhibit the converse or the inverse error. Use symbols to write the logical form of each argument. If the argument is valid, identify the rule of inference that guarantes its validity. Otherwise, state whether the converse or the inverse error is made. If I go to the movies. I won’t finish my homework. If I don’t finish my homework. I won’t do well on the exam tomorrow 3 If I go to the movies, I won’t do well on the exam tomorrow.27ES Some of the argents in 24-32 are valid. wherere as others ex the converse o the invene errot. Use symbols to wcite Vie Iogic.aI form of each arrieM. Il the argument is valid. ideify tie rt.Ie of irfeience that gu.arantcs its vidity. Otherwlie, state wt.ethei the converse or the inverse error is rade.Some of the arguments in 24-32 are valid, whereas others exhibit the converse or the inverse error. Use symbols to write the logical form of each argument. If the argument is valid, identify the rule of inference that guarantees its validity. Otherwise, state whether the converse or the inverse error is made. 29. If at least one of these two numbers is divisible by 6. then the product of these two numbers is divisible by 6. Neither of these two numbers is divisible by 6. The product of these two numbers is not divisible by 6.Some of the arguments in 24-32 are valid, whereas others exhibit the converse or the inverse error. Use symbols to write the logical form of each argument. If the argument is valid. Otherwise, state the rise of Inference that guarantees its validity. Otherwise, state whether the converse or the inverse error is made. If the compiler program is correct, then the correct output when run with the test data my teacher gave me.Some of the arguments in 24-32 are valis, whereas others exhibit the converse or the inverse error. Use symbols to write the logical form of each argument. If the argument is valid idenify the rule of inference that argument validity. Otherwise, state whether the converse or the inverse error ismade. Sandraknows Java and Sandra knows C++. SandraknowsC++Some of the arguments in 24-32 are valid, whereas others exhibit the converse or the inverse error. Use symbols to write the logical form of each argument. If the argument is valid, identify the rule of inference that guarantees its validity. Otherwise, state whether the converse or the inverse error is made. If I get a Christmas bonus, I'll buy a stereo. If I sell my motorcycle, I'll buy a stereo. If I get a Christmas bonus or I sell my motorcycle, then 1,11 buy a stereo.Give an example (other then Example 2.3.11) of a valid argument with a false conclusion.Give an example (other than Example 2.3.12) of an invalid argument with a true conclusion.35ES Given the following information about a computer program, find the mistale in the program. There is an undeclared variable or there is a syntax error in the first five lines. If there is a syntax error in the first five lines, then there is a missing semicolon or a variable name is misspelled. There is not a missing semicolon. There is not a misspelled variable name.In the back of an old cupboard you discusser a note signed by a pirate famous for his bizarre sense of humor and love of logical puzzles. In the note he wrote deal he had hidden treasure somewhere on the property. He listed five true statements (a-e below) and challenged the reader to use them to figure out the location of the treasure. a. If this house is next to a lake, then the treasure is not m the kitchen. b. If the tree in the front yard is an elm, then the treasure is in the kitchen. c. This house is next to a lake. d. The tree in the front yard is an elm or the treasure is buried under the flagpole. If the tree in the back yard is an oak, then the treasure is in the garage. Where is the treasure hidden?38ES The famous detective Percule Hoirot was called in to solve a baffling murder mystery. He determined the fallowing facts. Lord Hazelton, the murdered man, was killed by a blow on the head with a brass candlestick. Either Lady Hazelton or a maid, Sara, was in the dining room at the time of the murder. If the cook was in the kitchen at the time of the murder, then the butler killed Lord Hazelton with a fatal dose of strychnine. If Lady Hazelton was in the dining room at the time of the murder, then the chauffeur killed Lord Hazelton. If the cook was not in the kitchen at the time of the murder, then Sara was not in the dining room when the murder was committed. If Sara was in the dining room at the time the murder was committed, then the w me steward killed Lord Hazelton. Is it possible for the detective to deduce the identity of the murderer from these facts? If so, who did murder Lord Hazelton? (Assume there was only one cause of death.)40ES In 41—44 a set a pren.sei and a conclusion arc given. Use the valid argument forms listed in Table 2.3.1to deduce the conclusion from the premises, giving a reason for each step as in Example 2.3.8. Assume all variables are statement variables. ~pqr s~q ~t pt ~pr~s ~q In 41-44 a set premises and a conclusion are given. Use the valid argument forms listed in Table 2.3.1 to deduce the conclusion from the premises, giving a reason for each step as in Example 2.3.8, Assume all variables are statement variables. pq qr pst ~r ~qus t In 41-44 a set premises and a conclusion are given. Use the valid argument forms listed in Table 2.3.1 to deduce the conclusion from the premises, giving a reason for each step as in Example 2.3.8, Assume all variables are statement variables. ~pr~s ts u~p ~w uw ~t In 41-44 a wt o premises and a conclusion are given. Use the valid argument forms listed in Table 2.3.1 to deduce the conclusion from the premises, giving a reason lot each step as in Example 2.3.8. Assume all variables are statement variables. pq rs ~s~t ~qs ~s ~pru wt uw The input/output table for a digital logic circuit is a table that shows _______The Boolean expression that corresponds to a digital logic circuit is ________3TY 4TY 5TY 6TY 1ES Give the output signals for the circuits in 1—4 if the input signals are as indicated.Give the output signals for the circuits in 1—4 if the input signals are as indicated. P=1,Q=0,R=0 Give the output signals for the circuits in 1-4 if the input signals are as indicated. Input signals p=0,Q=0,R=0 5ES 6ES 7ES In 5-8, write an input/output table for the circuit in the referenced exercise. Exercise 4 9ES In 9-12, find the Boolean expression that corresponds to the circuit in the referenced exercise. Exercise 2 11ES In 9-12, find the Boolean expression that corresponds to the circuit in the referenced exercise. Exercise 4 13ES Construct circuits for the Boolean expressions in 13-17. (PvQ)15ES 16ES 17ES For each of the tables in 18-21, construct (a) a Boolean expression having the given table as its truth table arid (b) a circuit having the given table its input/output table.For each of the tables in 18-21, construct (a) a Boolean expression having the given table as its truth table and (b) a circuit having the given table its input/output table.For each of the tables in 18-21, construct (a) a Boolean expression having the given table as its truth table and (b) a circuit having the given table as its input/output table. 20 For each of the tables in 18-21, construct (a) a Boolean expression having the given table as its truth table and (b) a circuit having the given table as its input/output table.Design a circuit to take input signals P,Q, and R and output a 1 if, and only if, P and Q have the same value and Q and R have opposite values.Design a circuit to take input signals P,Q, and R and output a 1 if, and only if, all three of P,Q, and R have the same value.The light in a classroom are controlled by two switches: one at the back of the room and one at the front. Moving either switch to the opposite position turns the lights off if they are on if they are off. Assume the lights off if they are on and on if the yare off Assume the lights have been installed so that when both switches are in the down installed so that when both switches are in the down position, the lights are off. Design a circuit to control the switches.An alarm system has three different control panels in three different locations. To enable the system, switches in at lest two are in the on position. If fewer then two are in the on position, the system is disables. Design a circuit to control the switches.Use the properties listed in Thearem 2.1.1 to to show that each pair of circuits in 26-29 have the same input/output table. (Find the Boolean expressions for the circuits and show that they are logically equivalent whien regarded when regarded as statement forms.)Use the properties listed in Theorem 2.1.1 to show that each pair of circuits in 26-29 have the same input/output table. (Find the Boolean expressions for the circuits and show that they are logically equivalent when regarded as statement forms.) 27.Use the properties kited in Theorem 2.1.1 to show that each pair of circuits in 26-29 have the same input/output table. (Find the Boolean expressions for the circuits and show that they are logically equivalent when regarded as statement forms.)29ES For the circuits corresponding to the Boolean expressions in each of 30 and 31 there is an equivalent circuit with at most two logic gates. Find such a Circuit. (PQ)(~PQ)(~P~Q)31ES The Boolean expression for the circuit in Example 2.4.5 is (PQR)(P~QR)(P~Q~R) (a disjunctive normal form). Find a circuit with at most three logic gates that is equivalent to this circuit.Show that for the Sheffer stroke |, PQ(PQ)(PQ). Use the results of Example 2.4.7 and part (a) above to write P(~QR) using only Sheffer strokes.Show that the following logical equivalences hold for the Peirce arrow, where PQ=~(PQ) ~P=PP PQ=(PQ)(PQ) PQ=(PP)(QQ) Write PQ using Peirce arrows only. Write PQ using Peirce arrows only.To represent a nonnegative integer in binary notation means to write it is as sum of products of the form ______, where_____2TY

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