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Proofs Worksheet

Prove that if P → Q and Q → R, then P → R

Question #1

This is a proof of the transitive property.
If P → Q and Q → R then if P is true, Q must be true by the first implication, and hence R must be true by the second implication. Therefore, P → R

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Prove that P ^ Q is logically equivalent to Q ^ P

Question #2

If P ^ Q is true, then both P and Q must be true, so Q ^ P is also true If Q ^ P is true, then both Q and P must be true, so P ^ Q is also true Hence, the two statements are logically equivalent

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Prove that P → Q is logically equivalent to ~P v Q

Question #3

If P → Q is true, then P is true, Q must be true, and if P is false, Q can be either true or false. This is the same as ~P v Q If ~PvQ is true, then either P is false, or Q is true. This is the same as P → Q.

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Prove that all combinations of truth tables are expressible

Conjunctive Normal Form, Disjunctive Normal Form

Question #4

If P → Q is true, then P is true, Q must be true, and if P is false, Q can be either true or false. This is the same as ~P v Q If ~PvQ is true, then either P is false, or Q is true. This is the same as P → Q.

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Prove that P ^ (Q v R) is logically equivalent to (P ^ Q) v (P ^ R)

Question #5

If P ^ (Q v R) is true, then P must be true, and either Q or R must be true. Thus, either P ^ Q or P ^ R must be true, so (P ^ Q) v (P ^ R) is true If (P ^ Q) v (P ^ R) is true, then P must be true, and either Q or R must be true. Thus P ^ (Q v R) is true.

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