Logical EquivalentsDate: 03/03/97 at 06:11:37 From: Lewis Columber Subject: Logical equivalents Prove or disprove: existential x P(x) and existential x Q(x) is logically equivalent to existential x (P(x) and Q(x)). We are to prove this using logical equivalences. Thanks, Lewis Date: 03/04/97 at 17:31:55 From: Doctor Daniel Subject: Re: Logical equivalents Hi Lewis, I'm going to use this notation for the backwards capital e: _E, and ^ for and. We're to prove or disprove for any proposition P and Q that: (_Ex P(x)) ^ (_Ex Q(x)) <=> (_Ex (P(x)^Q(x))) Well, let's see what both of these say: The first says that some x makes P true, and some x makes Q true. The second says that some x makes both P and Q true. So it seems unlikely that one directly is equivalent to the other. And here's a simple counterexample: Suppose P(x) is the proposition that x is even, and Q(x) is the proposition that x is odd. Then, clearly, _Ex P(x); that just says there are even numbers. And also, _Ex Q(x); that just says there are odd numbers. But it's not true that _Ex(P(x)^Q(x)); that would mean there were a number x that is both odd and even; we know that this is false. So the question is false. -Doctor Daniel, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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