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The question I'm stuck on is:

Let $p$ and $q$ be propositions. A compound proposition $f(p,q)$ is given by $(p\vee \neg q) \Rightarrow (p \Leftrightarrow q)$. Find two non-equivalent propositions $h_1(q)$ and $h_2(q)$ which depend only on $q$, such that both propositions $f(p,q)\vee h_1(q)$ and $f(p,q)\vee h_2(q)$ are tautologies.

I have completed the truth table for the original proposition and have ended up with (T F T T) with both $p$ and $q$; but I'm unsure how to find a proposition where only $q$ is a variable (from what I know there are $4$ nonequivalent propositions for a single variable, I'm just not sure how to combine this information, or whether its right).

Thanks.

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1 Answer

Let us first analyze $f(p,q)$. It says that if $p$ is true or if $q$ is false, then $p$ and $q$ are equivalent. Namely, if $p$ is true then $q$ is true, and if $q$ is false then $p$ is false.

When does that fail? It fails when $p$ is true but $q$ is false. So we need to find two propositions depending only on $q$ such that whenever $q$ is false, they are true.

We can take $h_1(q)=q\lor\lnot q$, which is a tautology, so $f(p,q)\lor h_1(q)$ is also a tautology; and we can take $h_2(q)=\lnot q$, which is true whenever $q$ is false, so again we have that $f(p,q)\lor h_2(q)$ is a tautology. it is clear that the two propositions are non-equivalent because one is always true, and the other can be false.

My general advice for these sort of questions is to understand what does the proposition (in this case $f(p,q)$) mean semantically, and identify when it is false. Then it is not hard to come up with inequivalent propositions which complement it.

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+1 For finding my brain. I could not read properly and thought $f$ was a free variable. :) –  AD. Oct 17 '12 at 11:52
    
@AD.: It was right under the bouncy castle, I saw it by luck! –  Asaf Karagila Oct 17 '12 at 11:52
    
Thanks! $\,\,\,$ –  AD. Oct 17 '12 at 11:59
    
thanks for your help! i think i was over-complicating it –  Z Oj Oct 17 '12 at 12:13
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