Consider the reduction from 3-CNF-SAT to SUBSET-SUM. For the formula ϕ = (x1 ˅ ¬x3 ˅ x4) ˄ (¬x1 ˅ x3 ˅ x4) ˄ (¬x1 ˅ x2 ˅ ¬x4), construct the following values in the set S: v1, v1’, v2, v2’, v3, v3’, v4, v4’. Note that the formula has four variables: x1, x2, x3, and x4. It has three clauses: C1 = (x1 ˅ ¬x3 ˅ x4), C2 = (¬x1 ˅ x3 ˅ x4), and C3 = (¬x1 ˅ x2 ˅ ¬x4).

Q: Prove or disprove the statement There is no set S such that P(S) is denumerable.

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Q: Suppose A and B are bounded and nonempty subsets of real numbers and 1 E R. In this case: sup(A U B)…

A: Let A,B not empty, bounded subsets of R .Prove thatsup(A∪B)=max{sup(A),sup(B)}. Let: ⇒x∈A∪B⇒x∈A or…

Q: Let A be a set with |A| = n. Using induction, prove that |P(A)| = 2^n .

A: To show that the cardinality of power set of A is 2n when given that cardinality of A is n.

Q: 5. (a) If r23 = 0, prove that R₁ (23) = (23) = 712 + 7³1 01-23 = (1 - 12 - 13). 07.

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Q: For each positive integer n, define the set A, {2'(2n- 1) : ieZ2o}. (a) Prove that P {A1,A2,...} is…

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Q: Let a and b be integers such that b measures a. (a) Prove that –b measures a.

A: Given: b measures a To proof: -b measures a

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Q: Prove: |P(A)|= 2" where |A| = n. (Here P(A) is the power set of the set A with n elements.)

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Q: A and B are finite sets Prove that |AxB| = |A||B|. %3D

A: Let A=1, 2, 3, B=a, b, c

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Q: For Aand B subsets of formal A x CBU C) is a se t x iprovide subset

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Q: Let S be the set of numbers of the form { ((-1)^(n)) (2n + 1)/(4n - 1) } where n is an element of…

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Q: 1. Find the sup and inf of the following sets. {1/n :n e N} • {2 – 1/n :n € N}. • {e";r € Q}.

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Q: Suppose f: N→A is surjective. Prove or disprove: A has the same cardinality as N.

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A: Prove that A and B have the same cardinality.

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A: Use the formula of permutation

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Q: 2. Prove thatN is the smallest inductive subset of R:

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Q: Prove that for all sets A and B and nEZ, n ≥ 1, the following holds: 72 11 (a) U(A\B) = (UA) \ B;…

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A: We have to check whether set A = B.

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Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.