Prove the statement; a – c is a divisor of ab + cd=→a – c is a divisor of ad + bc.

Q: . Rewrite the equality of ideals (r+ y/2) = (3 – 4/2) in Z[/2] in terms of elements.

A: We know that, A subring  B of a ring R is called a ideal of R if for every r∈R and for every b∈B,…

Q: Prove that (AXB)×C=B(A • C)-C(A • B).

A: The solution is given in the next step.

Q: Show that if AB = AC and A is nonsingular, thenB = C.

A: Given that  A is invertible matrix. It means inverse of A matrix is possible. A is non-singular…

Q: Let a, b, x, y ∈Zbe such that ax+by= 11. May we conclude that gcd(a, b) = 11?Why or why not? (Write…

A: Consider the given information:Let a,b,x,y∈ℤ be such that ax+ by=11.

Q: Prove that |ab|= |a | |b| for any numbers a and b.

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

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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: Prove that AABC A DEF. C=12. D2,4) E(4, 2) 6-14.214 A6 4)

A: Compare the lengths of both the triangles.

Q: Let M = {-x² + 2,x + 1, 2x}. %3D Which of the following statements is true about M?

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Q: determine which of the followingmatrices are nonsingular:

A: Given, 1322141-72 We say that a matrix is singular if its determinant is zero otherwise we say it is…

Q: Prove that for A # B, AB S AB S AB.

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Q: Prove that if a eZ then the only positive divisor of both a and a+1 is 1.

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Q: Prove the following statement: 2" ≥ 2n

A: We have to prove the following statement- 2n≥2n We will prove it using mathematical induction.

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: Prove that for all a, b, and c e Z, if a † (2bc+12), then a † (b – 2) or a ł (c + 3).

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Q: Apply any method to prove that if a|b and alc then a|(bc – 3c)

A: We will use direct proof method to prove the given statement

Q: Prove that (A-1)-1 = A. %3|

A: Let A be inverse of matrix A. Then, by definition We know that

Q: Prove that ∀a, b, k ∈ Z that ((4|k) ∧ (k = ab)) ⇒ (2|a ∨ 2|b)

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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 is isomorphic to Z2.

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Q: Show that SU2(C) is simply connected and SO3(R) = RP³.

A: Consider SU2C group  SU2=α-β¯βα: α,β∈ℂ,α2+β2=1 To show it is simply connected we will show its…

Q: If A and B are independent, prove that B and the complement of A are independent.

A: Given,A and B are independent events , so P(A∩B)=P(A)×P(B) .

Q: Complete the congruence statement. AJMK = A by H M %23 K J

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Q: Prove that A(BC) = (AB)C.

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Q: Show that ∗ : R × R → R given by         (a, b) → a + 4b2 is a binary operation.

A: Given for all real number a and b

Q: The transitive property of congruence states that if 2 A = ZB and z B C, then A C. O True O False

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Q: Let (a, b) = 1. Prove (a2, b2) = 1.

A: Let (a, b) = 1. Prove (a2, b2) = 1. Given (a , b) = 1. Which means a is relatively prime to b

Q: Let a, y E R. Prove that if 21a + 35y = 45, then a and y are not both integers.

A: An integer is colloquially defined as a number that can be written without a fractional component.…

Q: If R is a UFD and f(x) , g(x) are elements of R[x] , then c(fg) = c(f)c(g), do not copy paste…

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Q: Prove that 1 (c) 2F₁ (a, b; c; z)

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Q: Prove that OfVg.dr = ] (Vfx Vg) · N ds.

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Q: Let jEZ (a) Prove directly the statement : If 3j, then 3j²

A: NOTE: According to guideline answer of first three subpart can be given, for other please ask in a…

Q: Let a ER which expressions are part of the Borel o algebra, B(R) they are correct? and give the…

A: The main objective is to determine the correct option.

Q: 2. Prove that any two non-zero elements a, b in a P.I.D. R have a g.c.d. Further if d e R is a…

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Q: Prove the following statement: 2" ≥2n

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Q: 5) Let Re(z) and Im(z) denote respectively the real parts of z, show that:- a) z+2 = 2 Re b) z-z =…

A: Introduction: Complex numbers are those that are expressed as a+ib, where a,b are real numbers and I…

Q: p) Prove that if ac | bc and c +0 then a | b.

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Q: If a is divisible by b, then bq, for some integer q. a

A: See the detailed solution below.

Q: If a<b<c<d, then prove that for any real 2,

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Q: Prove that √2 is not an element of Q(?).

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Q: prove if a is a divisor of B and B is a divisor if c, then a is a divisor of c

A: Explanation of the solution is given below..

Q: Suppose that P ( A ) = 0.4,  P ( B ) = 0.5 and A , B are disjoint. Compute P ( A c ∪ B c ) .

A: From the provided information, P (A) = 0.4, P (B) = 0.5 The events A and B are disjoint.

Q: If A⋂B are independent, prove that Ac⋂B are independent.

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Q: Let A = 1 2 1 and v = If the entries of A and v belong to Z3, what is the third entry of Av? Answer:

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Q: Let a * b = 2ab be defined on Z*. 1. Is * Commutative? 2. Is * Associative? 3. What is the identity…

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Q: 2. Determine the greatest common divisor of a(x) = x³ – 2 and b(x) = x + 1 in Q[x] and write it as a…

A: Given the polynomials   ax=x3-2 and bx=x+1 to find gcd(a(x), b(x)) and write it as a linear…

Q: (b) Is T one-to-one, onto or neither.

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Q: Prove that if alb and alc, then alb² + 3c + 2°c.

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