1. Determine whether the following statements are true or false. Prove or dispro

Question

1. Determine whether the following statements are true or false. Prove or disprove each of the of statements:

(1)  (?a)(?b)(?c)[(a | bc ? a not divide b) =? a | c], in the universe of all integers.

(2)  (?a)(?b)(?c)(?d)[a /b greater than c/d =? ad less than bc], in the universe of all non-zero real numbers.

(3)  (?a)(?b)(?c)(?d)[(a | (b ? 2c) ? a | (c ? d)) =? a | (b ? 2d)], in the universe of all integers.

Hint: Provided a rigorous proof if a statement is true. Give a concrete counterexample if a integers.

2. Claim. For all sets A, B, C and D, if A ? B and C ? D then A ? C ? B ? D.

Explanation / Answer

(1) (?a)(?b)(?c)[(a | bc ? a not divide b) =? a | c], in the universe of all integers. T

(2) (?a)(?b)(?c)(?d)[a /b greater than c/d =? ad less than bc], in the universe of all non-zero real numbers. T

(3) (?a)(?b)(?c)(?d)[(a | (b ? 2c) ? a | (c ? d)) =? a | (b ? 2d)], in the universe of all integers. F

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