Prove that A C B if and only if B C Ā.

Q: Prove: | a | - | b | <= | a + b |

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

A: We need to prove that if gcda,b=1 then gcda+b,ab=1 Let us on the contrary assume that  gcda+b,ab=c…

Q: Prove that if m*( A)= 0, then m* ( A UB) = m*( B).

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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: An element x is in A ∪ B if, and only if, _____________________.

A: Known fact: A∪B=x | x∈A or x∈B

Q: 3. prove that: det (AB) = det (A) det (B) If a - (;). » - ( :) 2 -1 7 A = 3 2 4

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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: 4. Let a,b,c € Z where c | (a+b). Prove that gcd(a,c) = gcd(b,c).

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Q: 4 Show that a + B = B +a for all a, BE C.

A: We can write alpha and Beta in general form of element of complex number.

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: a) Prove that (A∪B)c=Ac∩Bc. b) Prove thatA∪(B∩C) = (A∪B)∩(A∪C). c) Prove that ifA⊆B, thenA∩C⊆B∩C.…

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

A: We have to prove that

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: If a, b and c≠0 are integers, then a|b if and only if ac|bc.

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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: Prove this: A≤B iff AnB=A

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Q: Let a, b € Z and both nonzero. Prove that gcd(a, b) x lcm(a, b) = |ab|.

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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 if ACØ then A = (Ø.

A: To prove the given statement.

Q: Prove the following theorems: 1. If a /b, then a / bc.

A: Suppose that a | b. If a|b, then by definition there exist an integer m such that b=ma. That is,…

Q: Prove that B \ A = B if and only if A ∩ B = ∅.

A: We will use definition  of sets for proving following question

Q: 2. Let A = and B = al. Show that det(A + B) = det A + det B if and only if a + d = 0.

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

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Q: discrete

A: In method of contradiction we start by assuming that given statement is false. Then prove that it is…

Q: :)- Let A = and B = d Show that det(A + B) = det A + det B if and only if a + d = 0. %3D

A: In matric addition, the sum of two matrices is determined by adding the corresponding terms of the…

Q: If gcd(a, c) = 1 and gcd(b, c) = 1, prove that gcd(ab, c) = 1.

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Q: 4. Prove or provide a counter example for the following: (AU B)nC = (AnC)U(BnC)

A: We need to prove that(A∪B)∩C=(A∩C)∪(B∩C)If we have to prove that(AUB)∩ C C C…

Q: if and only if |z| =3 vlgn

A: True

Q: Prove: | ab | = | a | * | b | for all a's and b's in the reals.

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Q: Prove that:  If 0 ≠ p ∈ Q and x ∉ Q, then (show that) a + x ∉ Q and ax ∉ q.

A: Given: Let a∈ℚ and x∉ℚ To show: a+x∉ℚax∉ℚ

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

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

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Q: Prove: If A ∪ C = B and A ∪ D = B and A ∩ C = ∅ and A ∩ D = ∅, then C=D.

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Q: Prove or disprove that if (A U B) C (An B), then A = B

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Q: Prove that P(EFC) = P(E) – P(EF). |

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Q: Prove: If a > b, then a > 1/2 ( a + b ) > b.

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Q: Let a, b,c € R4. Prove that c+a a+b b+c c+b a +c b+a Al

A: a,b,c∈ℝ+ab+bc+ca≥c+ac+b+a+ba+c+b+cb+a

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

A: The given problem is to prove the following condition for given equation a+b=c, where a,b,c belongs…

Q: Prove that if A C B then P(A) C P(B).

A: Proof:

Q: a Let A Show that A is nonsingular if and d only if ad – bc + 0.

A: Known fact: A matrix is said to be invertible (nonsingular) if its determinant is nonzero.

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

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Q: Prove: Let a,b,x and y be integers. If a|x and b|y then (ab)|(2xy).

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Q: How to solve this?

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Q: a. If A and B are independent, and A and B are independent, show that A and B are also independent.

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Q: D

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

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Q: 4. Let a,b,c E Z where c| (a+b). Prove that gcd(a,c) = gcd(b,c).

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