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Discrete Mathematics With ApplicationsShow that Q, the set of all rational numbers, is dense along the number line by showing that given any two rational line by showing that given any two rational numbers r 1 and r 2 with r 1 < r 2 , there exists a rational number x such that r 1 < x < r 2 .
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Show that Q, the set of all rational numbers, is dense along the number line by showing that given any two rational line by showing that given any two rational numbers r 1 and r 2 with r 1 < r 2 , there exists a rational number x such that r 1 < x < r 2 .

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Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
Publisher: Cengage Learning,
ISBN: 9781337694193
BuyFind

Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
Publisher: Cengage Learning,
ISBN: 9781337694193

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Chapter 7.4, Problem 17ES
Textbook Problem

Show that Q, the set of all rational numbers, is dense along the number line by showing that given any two rational line by showing that given any two rational numbers r 1 and r 2 with r 1 < r 2 , there exists a rational number x such that r 1 < x < r 2 .

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Chapter 7 Solutions

Discrete Mathematics With Applications
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Ch. 7.1 - Given a function f from a set X to a set Y, f(x)...Ch. 7.1 - Given a function f from a set X to a set Y, if...Ch. 7.1 - Given a sunction f from a set X to a set Y, the...Ch. 7.1 - Given a function f then a set X to a set Y, if...Ch. 7.1 - Given a function f from a set X to a set Y, if yY...Ch. 7.1 - Given functions f and g from a set X to a set Y....Ch. 7.1 - Given positive real numbers x and b with b1 ....Ch. 7.1 - Given a function f from a set X to a set Y and a...Ch. 7.1 - Given a function f from a set X to a set Y and a...Ch. 7.1 - Let X={l,3,5} and Y=a,b,c,d) . Define g:XY by the...
Ch. 7.1 - Let X={1,3,5} and Y={a,b,c,d}. Define g:XY by the...Ch. 7.1 - Indicate whether the statement in parts (a)-(d)...Ch. 7.1 - a. Find all function from X={a,b}toY={u,v} . b....Ch. 7.1 - Let Iz be the identity function defined on the set...Ch. 7.1 - Find function defined on the sdet of nonnegative...Ch. 7.1 - Let A={1,2,3,4,5} , and define a function F:P(A)Z...Ch. 7.1 - Let Js={0,1,2,3,4} , and define a function F:JsJs...Ch. 7.1 - Define a function S:Z+Z+ as follows: For each...Ch. 7.1 - Let D be the set of all finite subsets of positive...Ch. 7.1 - Define F:ZZZZ as follows: For every ordered pair...Ch. 7.1 - Let JS={0,1,2,3,4} ,and define G:JsJsJsJs as...Ch. 7.1 - Let Js={0,1,2,3,4} , and define functions f:JsJs...Ch. 7.1 - Define functions H and K from R to R by the...Ch. 7.1 - Let F and G be functions from the set of all real...Ch. 7.1 - Let F and G be functions from the set of all real...Ch. 7.1 - Use the definition of logarthum to fill in the...Ch. 7.1 - Find exact values for each of the following...Ch. 7.1 - Use the definition of logarithm to prove that for...Ch. 7.1 - Use the definition of logarithm to prove that for...Ch. 7.1 - If b is any positive real number with b1 and x is...Ch. 7.1 - Use the unique factorizations for the integers...Ch. 7.1 - If b and y are positive real numbers such that...Ch. 7.1 - If b and y are positivereal numbers such that...Ch. 7.1 - Let A={2,3,5} and B={x,y}. Let p1 and p2 be the...Ch. 7.1 - Observe that mod and div can be defined as...Ch. 7.1 - Let S be the set of all strings of as and bs....Ch. 7.1 - Consider the coding and decoding functions E and D...Ch. 7.1 - Consider the Hamming distance function defined in...Ch. 7.1 - Draw arrow diagram for the Boolean functions...Ch. 7.1 - Fill in the following table to show the values of...Ch. 7.1 - Cosider the three-place Boolean function f defined...Ch. 7.1 - Student A tries to define a function g:QZ by the...Ch. 7.1 - Student C tries to define a function h:QQ by the...Ch. 7.1 - Let U={1,2,3,4} . Student A tries to define a...Ch. 7.1 - Let V={1,2,3} . Student C tries to define a...Ch. 7.1 - On certain computers the integer data type goed...Ch. 7.1 - Let X={a,b,c} and Y={r,s,tu,v,w} , Define f:XY as...Ch. 7.1 - Let X={1,2,3,4} and Y={a,b,c,e} . Define g:XY as...Ch. 7.1 - Let X and Y be sets, let A and B be any subsets of...Ch. 7.1 - In 41-49 let X and Y be sets, let A and B be any...Ch. 7.1 - In 41-49 let X and Y be sets, let A and B be any...Ch. 7.1 - In 41-49 let X and Y be sets, let A and B be any...Ch. 7.1 - In 41-49 let X and Y be sets, let A and B be any...Ch. 7.1 - In 41-49 let X and Y be sets, let A and B be any...Ch. 7.1 - In 41-49 let X and Y be sets, let A and B be any...Ch. 7.1 - In 41-49 let X and Y be sets, let A and B be any...Ch. 7.1 - In 41-49 let X and Y be sets, let A and B be any...Ch. 7.1 - In 41-49 let X and Y be sets, let A and B be any...Ch. 7.1 - Given a set S and a subset A, the characteristic...Ch. 7.1 - Each of exercises 51-53 refers to the Euler phi...Ch. 7.1 - Each of exercises 51-53 refers to the Euler phi...Ch. 7.1 - Each of exercises 51-53 refers to the Euler phi...Ch. 7.2 - If F is a function from a set X to a set Y, then F...Ch. 7.2 - If F is a function from a set X to a set Y, then F...Ch. 7.2 - If F is a function from a set X to a set Y, then F...Ch. 7.2 - If F is a function from a set X to a set Y, then F...Ch. 7.2 - The following two statements are_______....Ch. 7.2 - Given a function F:XY where X is an infinite set,...Ch. 7.2 - Given a function F:XY where X is an infinite set,...Ch. 7.2 - Given a function F:XY , to prove that F is not one...Ch. 7.2 - Given a function F:XY , to prove that F is not...Ch. 7.2 - A one-to-one correspondence from a set X to a set...Ch. 7.2 - If F is a one-to-one correspondence from a set X...Ch. 7.2 - The definition of onr-to-one is stated in two...Ch. 7.2 - Fill in each blank with the word most or least. a....Ch. 7.2 - When asked to state the definition of one-to-one,...Ch. 7.2 - Let f:XY be a function. True or false? A...Ch. 7.2 - All but two of the following statements are...Ch. 7.2 - Let X={1,5,9} and Y={3,4,7} . a. Define f:XY by...Ch. 7.2 - Let X={a,b,c,d} and Y={e,f,g} . Define functions F...Ch. 7.2 - Let X={a,b,c} and Y={d,e,f,g} . Define functions H...Ch. 7.2 - Let X={1,2,3},Y={1,2,3,4} , and Z= {1,2} Define a...Ch. 7.2 - a. Define f:ZZ by the rule f(n)=2n, for every...Ch. 7.2 - Define F:ZZZZ as follows. For every ordered pair...Ch. 7.2 - a. Define F:ZZ by the rule F(n)=23n for each...Ch. 7.2 - a. Define H:RR by the rule H(x)=x2 , for each real...Ch. 7.2 - Explain the mistake in the following “proof.”...Ch. 7.2 - In each of 15-18 a function f is defined on a set...Ch. 7.2 - In each of 15-18 a function f is defined on a set...Ch. 7.2 - In each of 15-18 a function f is defined on a set...Ch. 7.2 - In each of 15-18 a function f is defined on a set...Ch. 7.2 - Referring to Example 7.2.3, assume that records...Ch. 7.2 - Define Floor: RZ by the formula Floor (x)=x , for...Ch. 7.2 - Let S be the set of all string of 0’s and 1’s, and...Ch. 7.2 - Let S be the set of all strings of 0’s and 1’s,...Ch. 7.2 - Define F:P({a,b,c})Z as follaws: For every A in...Ch. 7.2 - Les S be the set of all strings of a’s and b’s,...Ch. 7.2 - Let S be the et of all strings is a’s and b’s, and...Ch. 7.2 - Define S:Z+Z+ by the rule: For each integer n,...Ch. 7.2 - Let D be the set of all set of all finite subsets...Ch. 7.2 - Define G:RRRR as follows:...Ch. 7.2 - Define H:RRRR as follows: H(x,y)=(x+1,2y) for...Ch. 7.2 - Define J=QQR by the rule J(r,s)=r+2s for each...Ch. 7.2 - De?ne F:Z+Z+Z+ and G:Z+Z+Z+ as follows: For each...Ch. 7.2 - a. Is log827=log23? Why or why not? b. Is...Ch. 7.2 - The properties of logarithm established in 33-35...Ch. 7.2 - The properties of logarithm established in 33-35...Ch. 7.2 - The properties of logarithm established in 33-35...Ch. 7.2 - Exercise 36 and 37 use the following definition:...Ch. 7.2 - Exercise 36 and 37 use the following definition:...Ch. 7.2 - Exercises 38 and 39 use the following definition:...Ch. 7.2 - Exercises 38 and 39 use the following definition:...Ch. 7.2 - Suppose F:XY is one—to—one. a. Prove that for...Ch. 7.2 - Suppose F:XY is into. Prove that for every subset...Ch. 7.2 - Let X={a,b,c,d,e}and Y={s,tu,v,w}. In each of 42...Ch. 7.2 - Let X={a,b,c,d,e}and Y={s,tu,v,w}. In each of 42...Ch. 7.2 - In 44-55 indicate which of the function in the...Ch. 7.2 - In 44-55 indicate which of the function in the...Ch. 7.2 - In 44-55 indicate which of the function in the...Ch. 7.2 - In 44-55 indicate which of the function in the...Ch. 7.2 - In 44-55 indicate which of the function in the...Ch. 7.2 - In 44-55 indicate which of the function in the...Ch. 7.2 - In 44-55 indicate which of the function in the...Ch. 7.2 - In 44-55 indicate which of the function in the...Ch. 7.2 - In 44-55 indicate which of the functions in the...Ch. 7.2 - In 44-55 indicate which of the functions in the...Ch. 7.2 - In 44-55 indicate which of the functions in the...Ch. 7.2 - In 44-55 indicate which of the functions in the...Ch. 7.2 - In Example 7.2.8 a one-to-one correspondence was...Ch. 7.2 - Write a computer algorithm to check whether a...Ch. 7.2 - Write a computer algorithm to check whether a...Ch. 7.3 - If f is a function from X to Y’,g is a function...Ch. 7.3 - If f is a function from X to Y and Ix and Iy are...Ch. 7.3 - If f is a one-to=-one correspondence from X to Y....Ch. 7.3 - If f is a one-to-one function from X to Y and g is...Ch. 7.3 - If f is an onto function from X to Y and g is an...Ch. 7.3 - In each of 1 and 2, functions f and g are defined...Ch. 7.3 - In each of 1 and 2, functions f and g are defined...Ch. 7.3 - In 3 and 4, functions F and G are defined by...Ch. 7.3 - In 3 and 4, functions F and G are defined by...Ch. 7.3 - Define f:RR by the rule f(x)=x for every real...Ch. 7.3 - Define F:ZZ and G:ZZ . By the rules F(a)=7a and...Ch. 7.3 - Define L:ZZ and M:ZZ by the rules L(a)=a2 and...Ch. 7.3 - Let S be the set of all strings in a’s and b’s and...Ch. 7.3 - Define F:RR and G:RZ by the following formulas:...Ch. 7.3 - Define F:ZZ and G:ZZ by the rules F(n)=2n and...Ch. 7.3 - Define F:RR and G:RR by the rules F(n)=3x and...Ch. 7.3 - The functions of each pair in 12—14 are inverse to...Ch. 7.3 - G:R+R+ and G1:RR+ are defined by G(x)=x2andG1(x)=x...Ch. 7.3 - H and H-1 are both defined from R={1} to R-{1} by...Ch. 7.3 - Explain how it follows from the definition of...Ch. 7.3 - Prove Theorem 7.3.1(b): If f is any function from...Ch. 7.3 - Prove Theorem 7.3.2(b): If f:XY is a one-to-one...Ch. 7.3 - Suppose Y and Z are sets and g:YZ is a one-to-one...Ch. 7.3 - If + f:XY and g:YZ are functions and gf is...Ch. 7.3 - If f:XY and g:YZ are function and gf is onto, must...Ch. 7.3 - If f:XY and g:YZ are function and gf is...Ch. 7.3 - If f:XY and g:YZ are functions and gf is onto,...Ch. 7.3 - Let f:WZ,g:XY , and h:YZ be functions. Must...Ch. 7.3 - True or False? Given any set X and given any...Ch. 7.3 - True or False? Given any set X and given any...Ch. 7.3 - In 26 and 27 find (gf)1,g1,f1, and f1g1 , and...Ch. 7.3 - In 26 and 27 find (gf)1,g1,f1 , and f1g1 by the...Ch. 7.3 - Prove or given a counterexample: If f:XY and g:YX...Ch. 7.3 - Suppose f:XY and g:YZ are both one-to-one and...Ch. 7.3 - Let f:XY and g:YZ. Is the following property true...Ch. 7.4 - A set is finite if, and only if,________Ch. 7.4 - To prove that a set A has the same cardinality as...Ch. 7.4 - The reflexive property of cardinality says that...Ch. 7.4 - The symmetric property of cardinality says that...Ch. 7.4 - The transitive property of cardinality say that...Ch. 7.4 - A set called countably infinite if, and only...Ch. 7.4 - A set is called countable if, and only if,_______Ch. 7.4 - In each of the following, fill in the blank the...Ch. 7.4 - The cantor diagonalization process is used to...Ch. 7.4 - When asked what it means to say that set A has the...Ch. 7.4 - Show that “there are as many squares as there are...Ch. 7.4 - Let 3Z={nZn=3k,forsomeintegerk} . Prove that Z and...Ch. 7.4 - Let O be the set of all odd integers. Prove that O...Ch. 7.4 - Let 25Z be the set of all integers that are...Ch. 7.4 - Use the functions I and J defined in the paragraph...Ch. 7.4 - (a) Check that the formula for F given at the end...Ch. 7.4 - Use the result of exercise 3 to prove that 3Z is...Ch. 7.4 - Show that the set of all nonnegative integers is...Ch. 7.4 - In 10-14 s denotes the sets of real numbers...Ch. 7.4 - In 10-14 s denotes the sets of real numbers...Ch. 7.4 - In 10-14 S denotes the set of real numbers...Ch. 7.4 - In 10—14 S denotes the set of real numbers...Ch. 7.4 - In 10—14 S denotes the set of real numbers...Ch. 7.4 - Show that the set of all bit string (string of 0’s...Ch. 7.4 - Show that Q, that set of all rational numbers, is...Ch. 7.4 - Show that Q, the set of all rational numbers, is...Ch. 7.4 - Must the average of two irrational numbers always...Ch. 7.4 - Show that the set of all irrational numbers is...Ch. 7.4 - Give two examples of functions from Z to Z that...Ch. 7.4 - Give two examples of function from Z to Z that are...Ch. 7.4 - Define a function g:Z+Z+Z+ by the formula...Ch. 7.4 - âa. Explain how to use the following diagram to...Ch. 7.4 - Prove that the function H defined analytically in...Ch. 7.4 - Prove that 0.1999….=0.2Ch. 7.4 - Prove that any infinite set contain a countable...Ch. 7.4 - Prove that if A is any countably infinite set, B...Ch. 7.4 - Prove that a disjoint union of any finite set and...Ch. 7.4 - Prove that a union of any two countably infinite...Ch. 7.4 - Use the result of exercise 29 to prove that the...Ch. 7.4 - Use the results of exercise 28 and 29 to prove...Ch. 7.4 - Prove that ZZ , the Cartesian product of the set...Ch. 7.4 - Use the results of exercises 27, 31, and 32 to...Ch. 7.4 - Let P(s) be the set of all subsets of set S, and...Ch. 7.4 - Let S be a set and P(S) be the set of all subsets...Ch. 7.4 - `The Schroeder-Bernstein theorem states the...Ch. 7.4 - Prove that if A and B are any countably infinite...Ch. 7.4 - Suppose A1,A2,A3,.... is an infinite sequence of...

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