In terms of logic and truth tables why is it that the truth table for exclusive or is as follows:

Consider $P$ and $Q$. Let $P + Q$ denote exclusive or. Then if $P$ and $Q$ are both true or are both false then $P + Q$ is false. If one of them is true and one of them is false then $P + Q$ is true. By exclusive or I mean $P$ or $Q$ but not both. I have been trying to figure out why the truth table is the way it is. For example if $P$ is true and $Q$ is true then no matter what would it be true?

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    $\begingroup$ That's the definition of exclusive or: either one or the other, but not both. $\endgroup$ – Shady_arc May 21 '14 at 15:47
  • $\begingroup$ You said it yourself: if $P$ and $Q$ are both true, then $P+Q$ is false... $\endgroup$ – fkraiem May 21 '14 at 15:47
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    $\begingroup$ It seems that you understand everything that you want to understand, you just don't understand that you understand it. $\endgroup$ – Asaf Karagila May 21 '14 at 15:49
  • $\begingroup$ Exclusive or captures some of the ways we use the word or. For example, "I either stay home or go out", but I can't have both. $\endgroup$ – Rui Baptista May 23 '14 at 13:08

The exclusive or operation (often xor, also known as either-or) only produces a true value if exactly one of the two statements is true; therefore, it has to be the case that either $P \wedge -Q$ or $-P \wedge Q$ (that is, either $P$ is true and $Q$ is false or $P$ is false and $Q$ is true).

If $P\vee Q$, that is, if $P$ and $Q$ are both true, then $P\wedge Q$ - $P$ or $Q$ - is true, because at least one of $P,Q$ is true, but this is the or operation, not the xor (exclusive or) operation.


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