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This is more related to engineering but I am having difficulty to draw the connection.

I have the following predicate a$\to$(b$\to$c) and I would like to find P$_{(a=T)}$ $\oplus$ P$_{(a=F)}$

I am doing the following:

P$_{(a=T)}$=T$\to$(b$\to$c)=(b$\to$c)

P$_{(a=F)}$=F$\to$(b$\to$c)=(b$\to$c) $\lor$ $\lnot$(b$\to$c)

Finally I get T, which turned out to be wrong as the correct answer is b $\land$$\lnot$c

I tried to find reference that would help refresh my logic math memory but couldn't find any. The scope I need include similar operations done to $\leftarrow$$\rightarrow$ (couldn't find it in mathjax) and $\oplus$

Thanks

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1 Answer 1

up vote 1 down vote accepted

We have $P_{\{a = F\}} \equiv F \to (b \to c) \equiv T$ (ex falso quodlibet, a statement $p\to q$ is always true, is $p$ is false). So $P_{\{a = T\}}\oplus P_{\{a = F\}} \equiv P_{a = T} \oplus T = \neg P_{\{a = T\}}$. Now $P_{\{a = T\}}= b\to c \equiv \neg b \lor c$, and hence $$ P_{\{a = T\}}\oplus P_{\{a = F\}} \equiv \neg P_{\{a = T\}} \equiv \neg(\neg b \lor c) \equiv b \land \neg c. $$

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