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Currently I'm stuck on these two problems and currently have no clue on how I'd go about figuring them out. These were previous homework problems.

  1. $p \vee q \rightarrow r \Leftrightarrow (p \rightarrow r) \wedge (q \rightarrow r)$

  2. $(p \rightarrow q) \wedge (q \rightarrow r) \wedge q \Leftrightarrow p \wedge q \wedge r$

I could only get this far with both. How would you go about doing these? And how would I go about approaching these types of problems?

1. \begin{align*} p \vee q \rightarrow r &\Leftrightarrow (p \rightarrow r) \wedge (q \rightarrow r)\\ &\Leftrightarrow (\neg p \vee r) \wedge (\neg q \vee r) &\text{Law of Implication}\\ &\Leftrightarrow (\neg p \vee \neg q) \wedge r &\text{Distributive Laws} \end{align*}

  1. \begin{align*} &(p \rightarrow q) \wedge (q \rightarrow r) \wedge q \Leftrightarrow p \wedge q \wedge r\\ &(\neg p \vee q) \wedge (\neg q \vee r) \wedge p \text{ via Law of Implication} \end{align*}
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    For (1) you got stuck because you made a mistake in applying distributivity: $(a \vee b) \wedge (a \vee c) = a \vee (b \wedge c)$. – 2017-02-02
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    Ah, you're right. I figured out the first one with that. Thanks. But what about the second one? – 2017-02-02
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    For (2), apply $a \wedge (\neg a \vee b) = a \wedge b$. (It's a special case of distributivity.) – 2017-02-02
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    I'm not sure what you mean by that. Where would I do this? – 2017-02-02
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    You apply it twice, first to $p \wedge (\neg p \vee q)$ and then to $p \wedge (\neg p \vee r)$. – 2017-02-02
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    What rule is this? I'm not sure what you're talking about still. You might be jumping ahead of me. I've not heard of what ever rule you're using. I'd be more helpful if you work it out and explain the steps. I don't know any "special cases" of distributivity. – 2017-02-02
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    You know distributivity: $p \wedge (r \vee q) = (p \wedge r) \vee (p \wedge q)$. Now, suppose $r = \neg p$. What happens to $p \wedge r$? – 2017-02-02
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    Let us [continue this discussion in chat](http://chat.stackexchange.com/rooms/52911/discussion-between-xenorosth-and-fabio-somenzi). – 2017-02-02

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