2. Let A = {n? +n € Z:n e Z} and B = {r e Z : r³ – 3x³ + 2x = %3D O mod 5}. Prove or disprove: A C B.

Q: Prove the following fact: A" = 3"-1A. 1 1 1 1 1 1 1 1 1 A =

A: Given, A =111111111  We have to  prove An = 3n-1A

Q: If z| =5 and w =- z+5 then Re(w) is equal to

A: Introduction :

Q: Let x E Z and y E Z. Prove the following statement: If y = 2 mod 4, then xy 1 mod 4.

A:

Q: 4. In the figure, if AB = AC and BO = OC. Prove that ZB = C .C B

A: a). Given: AB=AC and BO=OC Prove: ∠B=∠C

Q: 6. let A 6 6 -3 -2 -1 -2 -5 -3 diagomol moatie over te Lield R? Oer C I Is A similar do a

A:

Q: 3n³ + 20n² + 5 is O(n³)

A: Given: The given expression is 3n3+20n2+5 is 0(n3),

Q: 5n² is O(n³)

A: Suppose fn and gn are positive functions. We say that fn∈Ogn if there exist positive constants c, n0…

Q: Show that for n, m e Z+, gcd(2" – 1,2 – 1) = 28cd(n,m) – 1.

A: To prove : gcd(2n-1, 2m-1)=2gcd(n,m)-1where n,m ∈ℤ+

Q: Let A = {r € Z | r mod 15 = 5} and B = {x € Z| r mod 3 = 2}. Give an outline of a proof that A C B,…

A: Suppose we have two non-empty sets A and B . If x∈A then x∈B for all x∈A then A⊆B . If x∈B then x∈A…

Q: The evaluation of S S 3y³e*ydxdy is e - 1 O

A: Given: ∫01∫0y2 3y3exydxdy

Q: What is the first step to prove that S, 2+4 + 6+...+ 2n = n(2 + 1)? n(n + 1)? O Verify that S, is…

A: Here we have to determine the first step to prove that,  Sn⇒2+4+6+..........+2n=n(n+1)

Q: Show that if m e Z+ and a e Z such that gcd(a, m) = 1 and ordma = m – 1, then m is prime.

A:

Q: Suppose E={O1,O2}, and F={O1,O3}F. If Pr[E]=0.6 and Pr[F]=0.61, wha

A: From the given information,   E={O1,O2}, and F={O1,O3}F. If Pr[E]=0.6 and Pr[F]=0.61   Here, the…

Q: If In z + In(x – 6) = In(5x), then r

A:

Q: In 7 9 6. 6 5 is equal to A In r + Blns where A = and where B

A:

Q: 2. Suppose A is an m x n matrir, x ɛ Rª and Ax = b. Then b is in 1) R" | 2) R™ 3) RmfnT 4) R™n

A: Given :  A is an mxn matrix , x∈Rn and Ax=b   To find :  b is in which space of the given spaces.

Q: (d) The apartment is hot if and only if the temperature is not 75º and the air conditioning is not…

A: Here given Statement in English convert them to symbolic form using standard notation

Q: If OE = OF, then AB 2 CD. %3D

A:

Q: If x = y (mod n) and u = v (mod n) then which of the following is true (a) x + u = y + v (mod n) (b)…

A: Given that x≡y (mod n) and u≡v (mod n)

Q: Show that if P(A;) = 1 for all i > 1, then P(N1 A¡) = 1.

A: We know that P(Ai) = 1 for i ≥1 Considering P(Ai), i=1,2,3,... to be independent events

Q: 6. Suppose (a, m) = 1. Prove that for any b, Σ an + b) 1 (т — 1). n<m

A: Given (a,m) = 1 We have to prove that for any b , ∑n≤man+bm = 12(m-1)

Q: (9) Determine (with proof) whether EH(d)o(d) = 0. d|N

A: The given statement is not true when N is odd

Q: Problem 1: Prove A = {m e Z: m mod 6 = 2} is a proper subset of B = {n EZ:n is even}.

A:

Q: Q3. Prove that if ax = ay (mod m) , then x = y (mod m/d) where d = gcd( m, a).

A: According to our policy at Bartleby, I can solve only one question. I am solving Question 3. Let's…

Q: Suppose that A ¥BSCN D and x E A. Use the method of "proof by contradiction" to show that if x e C,…

A: Given data : x∈Aandx∉C To show : x∈B

Q: 5.7.4. Show that the two formulas (r = y) and (S(r) = Š(y)) in N.

A: Hi, since there are 2 questions given, the first one is answered.  Given - The formula  x=y To prove…

Q: The system x E 3 mod 9, x E 2 mod 4, has O A. no solution. O B. Unique solution. O C. finitely many…

A: Introduction: The Chinese remainder theorem is very significant in solving a system of linear…

Q: 1D Klove mothe mohical Inoucton 1. Mothe motical Imohchon forallinteges 2-3 nln+1) nt)

A: let this a statement P(n) For P(1) THE BOTH SIDES OF THE STATEMENT WILL BE 11+1=11+1  so P(1) is…

Q: Let a e Z. Prove that if 7 | 2a, then 7 | a.

A:

Q: If (ar + br) : 8n and n+1 %3D k-1 (ak - br) +, then E ak = n+2 k-1 k-1 None 2.

A:

Q: Show that if A is n × n with n odd and skew symmetric, then det(A) = 0.

A: Given:     A is n×n with n odd and skew symmetric.

Q: تفاضل وتكامل )2( نظري 1 If pconverges, then p could be xP to Select one: а. 3/2 stion 1 b. 1 Oc. non…

A: We have to check

Q: Let M = {14*: k E Z*} and A = {7k:k E Z*}. Then ACM A=M MC A None of these

A:

Q: P= 5 mod 8 show Ps² +byd = I mad 3P has for in teger S,y. ho Solution

A: Given: p≡5mod8 To show that px2+6y2≡1mod(3p) has no solution for x,y ∈ℤ

Q: 6. Show directly that inf(B) = 1 for B {N =: 4+1} -

A: The given expression is,  B = {1+1/n : n belong to N}

Q: If k = 0, then cosh(kx) Scosnlax) ax = sinnk) + C Select one: O True O False

A: Do the integration using concept of hyperbolic function.

Q: Suppose b1 + *+ bm < n. Prove that b,!bm!< n!.

A:

Q: 1. For an = 5n2 + 9n3 + 1, an = n2, prove or disprove: rn = O(an) (n → o)?

A: We will use the standard definition.

Q: Give a direct proof of the claim: Vn E N, if the last two digits of n are divisible by 4 then n is…

A:

Q: Ifx= y (mod n) and u =v (mod n) then which of the following is false. (a) x+u= y+ v (mod n) (b) хи в…

A:

Q: 5. Let C = {x ∈ Z|x ≡ 7( mod 9)} and D = {x ∈ Z|x ≡ 1( mod 3)}. (a) If C ⊆ D? Justify your…

A:

Q: (a) Ha: B1 0, with n = 26 andt=1.91 <p< 1 (b) Ha: B1 # 0, with n = 12, b1 = 0.16, and sp1 = 0.06 %3D…

A:

Q: . Let, a, = 1, az = 2, b; = 0, b2 = 1, for n2 3, a, = 2an-1 + bn-1 for n22 b, = br-1+ an-1 express…

A:

Q: If сx 3D су (тod n) and c >0 then а) х%3D у (тоd cn) b) х%3D у (тod n) c) x = y (mod“) d) None of…

A: We have to solve given problem:

Q: Show that if A and B are both n × n, then (a) det(AT BT ) = det(A) det(BT ); (b) det(AT BT ) =…

A: Given A and B are n×n matrix.

Q: If cx = cy (mod n) and c > 0 then b) х%3D у (тоd n) c) x= y (mod") d) а) х у (тod cn) None of these…

A: Congruence equation

Q: Q3. Prove that if kx = ky (mod m) , then x = y (mod m/d) where d = gcd( m, k).

A:

Q: Q1: Prove that ; If CE AB, D E AB and C# D, then CD = 2 %3D

A: Key point: 1. There is only one line possible from two distinct points.2. The line is expressed in…

Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.