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

Q: Given: Z1 = 22 АВ ВЕ Prove: АС CD A

A: Definition used -            Vertically opposite angles are equal in measure.             AA…

Q: 4. Complete the following proof in the format of your choice. D. 3. Given: AB DC and AB || DC Prove:…

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: Verify that for all x, the change-of-base formula holds: bx = ax Ioga b for a, b > 0 such that a…

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: 1. In the diagram below, BFCE, ABIBE DE 1 BE, and ZBFD = ZECA. Prove that AABC ADEF. D. B CE

A: Given,  AB⊥BE , DE⊥BE and ∠BFD≅∠ECA To prove ∆ABC~∆DEF In triangle ∆ABC and ∆DEF AB⊥BE and DE⊥BE…

Q: Is A – (B U C)= (A – B) N (A – C)? Prove it or disprove it.

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Q: If a+a e c(R) for all ae R. then Commutative is prove that

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Q: Let n E Z. Prove that if 3/ (2n² + 1), then 3 | r n.

A: Solution:     Let 3 does not divide n      => n = 3k ± 1 for some integer k.       => 2n2 + 1…

Q: Prove Theorem A(rB) = r (AB) = (rA)B

A: Given: A(rB) = r (AB) = (rA)B

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

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

Q: Let AB = AC. Prove that if det(A) = 0, then B = C.

A: The given question Let AB = AC. Prove that if det(A) = 0, then B = C. this question is not correct…

Q: Prove that the following relationship is true: nCr + nCr + 1 = n + 1Cr + 1

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Q: Prove that la + b + c| < ]a] + ]b| + |c[

A: Prove that,    a+b+c≤a+b+c

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

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Q: Let a e Z. Prove that if 6 | a and 10 | a, then 15 | a.

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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: 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: Prove that if T+¬AFB and I+¬A¬B, then I EA

A: Given proposition is Γ+¬A-B and Γ+¬A-¬B To prove Γ-A Now,  1. Γ+¬A-B∧Γ+¬A-¬B…

Q: BC = a. Prove: AB = 2a and AC = aV3. %3D %3D

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Q: Prove that A CB P(A) C P(B).

A: We have to prove that

Q: Given: AB s BC, AF = KC, MZDKA=mZEFC Prove: AD = EC В D А F K C er: mZA = m2 by reason AADK = A by…

A: topic - congruence of triangles

Q: Show that if AB = I, then det(A) 0 and det(B) # 0.

A: Here given as AB=In find determinant both sides detAB=detIndetAB=1

Q: Prove this: A≤B iff AnB=A

A:

Q: Prove that if A ∈ Mn×n(F) has two identical columns, then det(A) = 0.

A: The determinant is the scalar value computed by expanding a row or a column.

Q: Let AB = AC. Prove that if det(A) + 0, then B = C.

A: Given: Let AB=AC. To Prove: If detA≠0, then B=C

Q: PROVE that 2a + b = 0.

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Q: Prove that 223 a E lo,1) く 22 + 1영 ( 1-계 3(1-x2)

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Q: prove that n E vol(B6)) > II(b: – a;) vol( B jEJ i=1

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Q: Let A, B, and C e M,(C) such that ВС - СВ %3D А"А prove that BC = CB.

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Q: nd C such that AC=E

A:

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: 2) Prove the following identities: (a) A\B = A\(A O B); (b) A o (B\C) = (A O B)\CA Q C) = (A O B)\C.

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Q: .Prove that la + b + c| < |a] + |b| + ]c]

A:

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: Prove: (i) b (A) CA if and only if A is closed. (ii) b(A) n A 9 if and only if A is open. (li1) b(A)…

A:

Q: Prove Theorem . k(AB) = {kA)B = A{kB)

A: Proof: Given:   Let A and B are matrices of the same order and let k be the scalar.  To prove:…

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

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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 that there exist c1, c2 ER such that for all ne N C1 E1 sin (i)<c2 =D1

A: We will use the range of the sine function to determine the real constants c1 and c2

Q: Prove: For c &gt;= 0 , | a | &lt;= c if and only if -c &lt;= a &lt;= c

A:

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

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Q: Prove that for а + —1,а > 0, пENis (1+ а)" > 1+ па.

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

A: solution.

Q: prove that a(a+b)=c(c+d). a b d

A:

Q: Prove that A + (B + C) = (A +B) + C P, A+B

A:

Q: By using Caushy Rieman prove that (2) and (3) (2) 4ca) = e (3) .w prove Thot()-

A: (a)

Q: Prove this: A=B iff A° U B = Ω =

A: Solution : First prove left to right, If A⊂B is true , All elements of A are inside B, Then Ac is…

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: Prove that if A,B ∈Mn×n(F) are similar, then det(A) = det(B).

A: If the matrices A and B are similar then there exists a non singular matrix S,