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3. (4 points each) Prove or disprove: (a) f : Z Z X Z defin...

3. (4 points each) Prove or disprove: (a) f : Z Z X Z defined by f(n) = (n + 1, 2n) for any 72 € Z is a surjective function. 6. (8 points) Let X be a subset of N14. Prove that if IX] = 8, then there exist disjoint subsets A, B of X both of cardinality 2 such that the sum of the elements in A...

Exercise 4-11 Prove that the following arguments are valid....

Exercise 4-11 Prove that the following arguments are valid. These proofs especially emphasize Dist, Comm, and Assoc. This exercise is fairly challenging. Remember that Dist, like all our equivalence rules, works in both directions. (7) 1. (A . B) V (C. D) /.. (C- D) V A (9) 1. [(A- B) . D] V (...

I. Prove valid (. (x)(Sx v-ix) (Fx-Px 2. (y) lays-Py) 3...

I. Prove valid (. (x)(Sx v-ix) (Fx-Px 2. (y) lays-Py) 3. (3nt-Qx2 RX) 4. (Y)C-Ryv-5y) 2. Prove valid 1. (z) (PZ)QZ) (3ri((-yyv--o) 2. 3. (x) (-Px vox) 4 (x) 3 Prove invalid You must we the method of interpretation (Fx)(Ax B x) (x) (-BXV-Cx) - (x) (-Axv-Cx) 4. Prove valid (IN(-H...

A. True False (1 ot each) 1. The following formula is a wel...

A. True False (1 ot each) 1. The following formula is a well-formed formula P&Q&R 2. A valid argument can have false premises and true conclusion. 3. The formulas that make up disjunction are called conjuncts. 4. The formal proof system allows us to determine the validity and in...

[C Prove the validity of the following arguments using the m...

[C Prove the validity of the following arguments using the method of natural deduction. 60 pts total. (1) 1.A 2.-A .B (2) I.AbB 2.B5C 3.CCD A5D (3) 1. ~A 2. AvB i-AoB (4) 1. AeB 2.B5C .A.C (5) 1. ~(A+B) 2. C=(A+B) 3.-C5-D - in-D (6) 1. - ~C5(AvB) is 2. ~B 3.-B2-C Ae~B + (7) ...

Logic A. True or False (1 pt each) 1. The following formul...

Logic A. True or False (1 pt each) 1. The following formula is a well-formed formula P&Q&R 2. A valid argument can have false premises and a true conclusion. 3. The formulas that make up a disjunction are called conjuncts. 4. The formal proof system allows us to determine the va...

2 Sets, partial orders, and graphs Problem 2.1. Let bea tha...

2 Sets, partial orders, and graphs Problem 2.1. Let bea that closed under union that that ifAEMandBe then AUB € M. Prove the following statement by mathematica induction M, for all natural Problem 2.2 Let and let L be A*, thati the set all strings over Let y mean that string sel such that ...

n ) For any nonnegative integer n, show that = i n E(:...

n ) For any nonnegative integer n, show that = i n E(:)-E(%) n ) For any positive integer n, prove that Hint for one way of proving this: If X is set of cardinality n. find i is even i is odd a bijection between the set of elements of P(X) of even cardinality and the set of elements of ...

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