Let a; b; c 2 N such that a + b = c. Prove that gcd(a; b) = gcd(a; c) = gcd(b; c).

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 For all A ∈ A, we have 0≤ P(A) ≤ 1

A: Concept: Probability: Probability signifies the possibility of something happening. It's a…

Q: (a) For all a, b e Z prove that a³b² – ab* is even.

A: Properties of algebra Addition of two odd number is even  Addition of two even number is even…

Q: Let A = {4n^2 + 1|n ∈ Z}. Show that A is a subset of Z but not equal.

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

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Q: Show that if gcd (a, y) = 1 and a \ xy Then a\ x

A: Definition : Greatest common divisor ( gcd): Let  a and b be are two  integers, not both 0.  A…

Q: Let p». Prove that, for every e > 0, a.a.s. the largest component of G(n, p) has size at least…

A:

Q: 2. Let A € M₂ (R) be given. (a) If AA¹ = 0, prove that A = On. Does this hold if A € M₁ (C)? (b)…

A: This is a problem of Linear Algebra.

Q: Prove that S4 is not isomorphic to D12.

A:

Q: Let A € M₂ (R) be given. (a) If AA¹ = 0n, prove that A = On. Does this hold if A € M₂(C)? (b)…

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

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Q: 1. Prove that if A is infinite and a E A, then A – {a} is also infinite.

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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: Prove that det(AB) = det(A)· det(B) for any A,B ∈M2×2(F).

A: Let the matrix A and B as follows.

Q: Let g(x) and h(x) belong to Z[x] and let h(x) be monic. If h(x) divides g(x) in Q[x], show that h(x)…

A: Given: Let's take g(x) and h(x) belongs to Z(x) and take h(x) as monic.

Q: If E, and E2 are measurable then (E,UE,) is not measurable .

A:

Q: If X has full rank, so that P = X(X'X)-'X', prove that C(P) = C(X).

A: Given information: It is given that X is a full rank matrix.

Q: If E, and E2 are measurable then (E,UE) is not measurable .

A:

Q: Given an n x n-matriz A over a endent iff det(A) = det(A¹,...,A") =

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Q: locally Lipschitz in (t,x)E[a, b]x D.

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Q: If x is a lower bound for S in R then x must be unique..

A: If x is a lower bound for S in R then x must be unique.

Q: Show that the Modulus Function            f : R → R, given by f(x) = | x |, is neither one-one nor…

A: Consider, f: R → R given by f(x) = | x |Where,| x | is x, if x is positive or 0| x | is – x, if x is…

Q: 1. Prove that if A is infinite and a e A, then A – {a} is also - infinite.

A:

Q: Show that if gcd (a, y) = 1 and a \ xy Then a \ x

A: if 1. gcd(a,b)=1then there exist some integers m and n such that am+bn=12. suppose a and b are…

Q: Prove that all elements in S are multiples of 3. by Structualindrction 3ES X,Y ES > (X y )ES

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Q: Prove that if x is not free in the formula A, then ` A ≡ (∃x)A

A: The formula is -------(1). To Prove: x is not free in the formula Proof: Let us assume that A(x) :…

Q: Let G=(V, E) be a finite

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

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Q: = Prove that E = ehd.

A: We know that Ef(x)=f(x+h)

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: Let x, y, A, u E Z. Prove that if 1 = Ax + µy then r, y are relatively prime.

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Q: Prove that if gcd(a, b) = 1 and cla, then gcd(b, c) = 1.

A: We need to prove If gcd(a,b)=1, then gcd(b,c)=1. The proof is given below.

Q: Show that if gcd (a, y) = 1 and a \xy Then a\ x

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Q: If x]y then gcd (x, y) = y. إختر واحدا: Las O

A: Given x|y hence y=ax It implies that y is a multiple of x

Q: Prove Theorem  A + (B + C ) = (A + B ) + C .

A:

Q: Given: mAB = x Prove: mz1 = 180° - x A D Exercises 25, 26

A: Given:           m AB⏜ = x   To prove:              m<1 = 1800 - x

Q: Show that cl(a) = {a} if and only if a ∈ Z(G).

A:

Q: Let i, r, and k be positive integers, and let s gcd(i, r). Assume that r | ik. Prove that | k.

A: To prove the divisibility property under the given conditions

Q: Let a ∈ G be such that |a| = ∞ . For m≠n prove that a^m ≠a^n

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Q: Prove for a e Z and n e N that gcd(a, a + n) | n.

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Q: Please solve...

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Q: Show that |Re(z)|+|Im(z)| < v2|z| for any z e C.

A: The solution is given as

Q: Let P = {X E P(Z*)|X is finite}. Prove that P is denumerable.

A:

Q: Let d = gcd(a, b). If a = da' and b = db', show that gcd(a', b') = 1.

A: Let gcd(a'b')=1 since d= gcd(a,b), there are integers p,q which satisfy the equation as: pa+qb=d

Q: 4) Let m,n, r E Z with m # 0 + n. Show that gcd(m, n) divides r if and only if there exist a, b E Z…

A: Let gcd(m,n) divides r. Then we have to prove that, a·m+b·n=r Since, gcd(m,n) divides r than by…

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

A: solution.

Q: Let a, b ∈ Z, not both zero. Prove gcd(−a, b) = gcd(a,−b).

A: We have gcd(-a , b)=gcd(a,-b) let gcd(-a , b)=d Where -a=de  , b=df  ⇒a=d(-e) , -b=d(-f)⇒d is…

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 for any k > 0, nk e o(2v™).

A: The complete solutions are given below