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Question: Let $P_1,\ldots,P_n$ be propositional variables. When is the statement $P_1 \oplus \cdots \oplus P_n$ true?

I'm currently learning the basics of discrete math. I am stuck on this last question of my assignment... not really sure how to go about solving it.

I do know that a propositional variable can either be true or false.

Thanks

Michael Hardy
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I'm Hungry
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2 Answers2

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Idea: You may think of $\oplus$ as acting like mod $2$ addition by taking $T$ as $1$ and $F$ as $0$. Then \begin{align*} T \oplus F &=1+0=1 \pmod{2}\\ T \oplus T &=1+1=0 \pmod{2}\\ F \oplus F &=0+0=0 \pmod{2} \end{align*}

Let $k$ be the number of statements that are true among $P_1, P_2, \ldots ,P_n$, so the remaining $n-k$ are false. Then $\oplus_{i=1}^{n}P_i$ is true if and only if $k$ is odd because based on the idea I have suggested above, we can think of $\oplus_{i=1}^{n}P_{i}$ as $\underbrace{1+1+\dotsb+1}_{k} \equiv k \pmod{2}.$

Anurag A
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Hint: $x \bigoplus {\rm False}$ has the same truth value as $x$.

$x \bigoplus {\rm True}$ has the opposite truth value as $x$.

So every time you have a True among the arguments, the truth value flips.

paw88789
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