Prove or give a counterexample to each of the following: (1) (A1 x A2) U (BỊ × B2) = (A1 U B1) × (A2 U B2) for all sets Aj, A2, B1, B2. (2) (A1 x A2) n (Bị x B2) = (A1 n B1) x (A2N B2) for all sets A1, A2, B1, B2.

Q: 1: Prove that (c a) * H = (c b) * H implies that a H = b H

A: Given an equation. We need to prove that : a*H = b*H where H is a subgroup of a group ( say G). Also…

Q: Prove that 1/r + 1/r2 + 1/r3 + . . . = 1/r − 1, for |r|  > 1.

A: we have to prove that: 1r+1r2+1r3+. . . . . .=1r-1  ,  for r>1

Q: Prove that: 1- Ā. (B × C) = B . (Č × Ā) = C . (Ā × B)

A: As per our guideline we are supposed to answer only one asked question.kindly repost other question.…

Q: Show that U(8) and U(10) are not isomorphic.

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Q: Prove that det(AB) = det(A) · det(B) for any A, Bϵ M2x2(F).

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Q: verify that det(AB) = det(BA) and deter- mine whether the equality det(A + B) = det(A) + det(B)…

A: See the attachment

Q: Prove that if gcd(a, b) = 1, then gcd(a + b, ab) = 1.

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Q: So then, if Az = h and Ay = V26 + h) –2 – V10 Ay Compute and reduce

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Q: Prove that if A2 = A, then I − 2A = (I − 2A)−1.

A: Let A2=A To prove :  I − 2A = (I − 2A)−1

Q: 4. Prove that (À x B) - (B x C) x (C x A) = [AB %3D

A: The solution is given below.

Q: Prove that if both y and z are solutions to x' = Ax, then for any c1, c2, the function c1y + c2z is…

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Q: 5. Prove that arg(z1zz) = argzi – argzz %3D

A: Introduction: For a complex number z=x+iy, where x is the real part and y is the imaginary part,…

Q: Prove that 1/ r + 1/ r2 + 1/ r3 + . . . = 1/ r−1 , for ∣r∣ > 1.

A: Consider, It is a geometric series with,

Q: Prove that (a; + b;) = a; + i= 1 Σύ. i=1 i=1

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Q: Prove that if x is even and y is even, then the product of x and y is divisible by 4.

A: We use the Division Algorithm to prove the given statement.

Q: HeW. By the Same way Prove that: X Ju (x) (X)

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Q: Prove that if gcd (a,b)=1>then gcd (ab,ab)=l

A: I have proved using the fact that gcd can be expressed as integers linear combination q

Q: Let A = {n EZ | n = 5r for some integer r} and B = {mEZ | m = 20s for some integer s}. Determine…

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Q: Prove that Σ = 2" . a. x=0 b. = 0. x=0

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Q: Prove that if m is a positive interger, l'(-m+) = 1.3.5..(2m -1)

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Q: Show that (AB)¹ = BTA¹ if A = 2 0 2 0 and B = -1 1 15) -3 1 -1 -1 2 2). a

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Q: 2. Prove that if 5BA+ ACA – I = A, then A must be invertible.

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Q: Prove that |A U B| + |A n B| = |A| + |B|

A: Let A and B are two sets. A = number of elements in the set A. B = number of elements in the set B.…

Q: Prove that there are integers x and y such that 10x-13y=1

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Q: Show that U(8) is not isomorphic to U(10).

A: To Show: U(8) is not isomorphic to U(10)

Q: Prove that X and Y are independent if and only if R = 0

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Q: 1. Given: mZABC 80° |3D Prove: x = 8 Statements 6x+5 3x +3 Reasons MLABC 80 MLABD + MLDBC MLABC (3x+…

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Q: Prove that 1 (c) 2F₁ (a, b; c; z)

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Q: Prove directly that no number of the form 4a (8b − 1) can be written as the sum of three squares by…

A: Let us first show that (8b-1) cannot be expressed as a sum of three squares Let us take any integer…

Q: Prove that there do not exist a, b ∈ Z satisfying 54a − 39b = 2.

A: Solution:

Q: Prove that if A ϵ Mnxn(F) has two identical columns, then det(A) = 0.

A: Given: A∈Mn×nF i.e. A is a matrix of order n×n.   Consider a matrix with first two column as…

Q: 6. Prove that if gcd(a, b) 1, then gcd(a +b, ab) = 1. %3D %3D

A: Given that gcd(a,b)=1 If possible , let us assume that  gcd(a+b,ab)=d,d≠1 Then the number d must…

Q: Prove: Provided it exists, (A-1)¯ = A.

A: Hey, since there are multiple questions posted, we will answer first question. If you want any…

Q: if z1-z2 and z1/z2 are both real then prove that two must either real or equal to each other

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Q: Prove that: (1+ x)*** (n+ 1)x C. + + 2 3 n+1

A: I am going to solve the given problem by using some simple calculus to get the required result.

Q: Prove that if gcd(a, b) = 1 and gcd(a, c) = 1, then gcd(a, bc) = 1. %3D

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Q: 3. Construct a bijection f: [0, 1] → (0, 1) UN.

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Q: Prove: For c >= 0 , | a | <= c if and only if -c <= a <= c

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Q: Prove that Z Q has the universal property of inverting non-zero elements.

A: in Mathematics, Z is the set of Integers. and Q is the set of all rational numbers. Z→Q Let f(α)=1α…

Q: Prove that |z₁+z₂²2 = |z₁|² + 12₂², if z₁/z₂ is purely imaginary.

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Q: Prove that −(−v) =v

A: To prove: --v=v

Q: True or False? If d = gcd(a,b), then there exists integers x, y such that d = ax +by

A: solution.

Q: 6x-C), Prove that If2zf(x+ct)+Ø て

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Q: Prove that (aA + bB)t = aAt + bBt for any A , B E Mmxn(F) and any a,bϵ F.

A: Consider, aA+bBt=aAt+bBt    ∵A+Bt=At+Bt=aAt+bBt     ∵kAt=kAt for any scalar k.=aAt+bBt

Q: Prove that: ā x (B xẽ) = (ã č)b – (ā-B )ë |

A: We have to prove that a→×b→×c→=a→⋅c→b→−a→⋅b→c→. Let a→=a1i^+a2j^+a3k^ b→=b1i^+b2j^+b3k^ And…

Q: Under what conditions on A and B is (AB)-1 = A-1B-1? Prove your assertion

A: Assertion: If A and B are commutative, then  AB-1= A-1B-1.

Q: 4. Verify thatf (x) = x³ – x2 – 20x + 6 satisfies the three conditions of Rolle's Theorem on [0,5].…

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