Is it true that
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p->(r ∧ q) is logically equivalent to (p->r) ∧ (p->q),
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p->(r ∨ q) is logically equivalent to (p->r) ∨ (p->q)?
I am solving problems on logical arguments (check whether they are valid) and it would be easier if I could break down the statements above and check them seperately. I wrote the truth tables, which turned out to match, so I think they should be equivalent, but I am not sure.
Below are the truth tables that I used
Best Answer
You can also apply the properties of the logical operators involved: \begin{align*} (p\to(q\wedge r)) & \Longleftrightarrow (\neg p\vee(q\wedge r))\\\\ & \Longleftrightarrow (\neg p\vee q)\wedge(\neg p\vee r)\\\\ & \Longleftrightarrow (p\to q)\wedge(p\to r) \end{align*}
Similarly, we do also have that \begin{align*} (p\to(q\vee r)) & \Longleftrightarrow (\neg p\vee (q\vee r))\\\\ & \Longleftrightarrow ((\neg p\vee\neg p)\vee(q\vee r))\\\\ & \Longleftrightarrow (\neg p\vee q)\vee(\neg p\vee r)\\\\ & \Longleftrightarrow (p\to q)\vee(p\to r) \end{align*} and we are done.