I know that the truth tables for both $p \to q$ and $p \wedge q$ are both different, but I cannot however figure out a statement that would make one true but the other false. If any would could help with this I would be very appreciative!
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1What is x, what is p, what is q...?? – Timbuc Sep 26 '14 at 17:34
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Try out every combination of x, p and q being true or false and see what the result is. There's only 8 ways. – Milo Brandt Sep 26 '14 at 17:36
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1What is this "x implies q and p and q both different"? – Git Gud Sep 26 '14 at 17:36
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1@Josh: I interpret your question as you wish to know about "the relation" between $p\rightarrow q$ and $p \wedge q$..? – Lehs Sep 26 '14 at 17:45
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@Lehs yes you are correct I just changed it – Sep 26 '14 at 17:50
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See this question to verify the truth table for $p\to q$ – cjferes Sep 26 '14 at 18:00
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It seems like Josh has been deleted. – Lehs Sep 26 '14 at 18:33
3 Answers
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When $P$ is false, $P\implies Q$ is true but $P\land Q$ is false.
Let $P$ be your favorite false proposition. $Q$ can be any proposition.
James Holbert
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One big difference between $p\rightarrow q$ and $p \wedge q$ is that the former always is true when $p$ is false, while the latter always is false when $p$ is false. These kind of formulas has one foot in human language and the other in algebra. Sometimes this is confusing for beginners.
An example then:
If it is easy to prove my tautology, then someone will do it.
It is easy to prove my tautology and someone will do it.
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Set $p = F$ and $q = T$. $p \Rightarrow q $ is therefore true in this instance, yet $p\wedge q$ would be false. Remember $(F \Rightarrow T) = T$, and $(F \wedge T) = F$, from the truth tables
