Prove that the proposition $p\to(q\to(p\wedge q))$ is a tautology by using identities.

Question

This is a homework question that I am struggling with.

I can use properties to expand out or simplify this proposition, but how does one prove that it is a tautology without using a truth table or some other form of concrete values? I have proven that it is a tautology using truth tables, but I have no idea how to even start doing this using expressions and identities.

Any insight would be much appreciated.

score 1 · Accepted Answer · answered Jan 16 '22 at 23:00

1

We can rearrange the proposed propositional expression as follows

\begin{align*} p\to(q\to(p\wedge q)) & \Longleftrightarrow \neg p\vee(q\to(p\wedge q))\\\\ & \Longleftrightarrow \neg p\vee(\neg q\vee(p\wedge q))\\\\ & \Longleftrightarrow (\neg p\vee\neg q)\vee(p\wedge q)\\\\ & \Longleftrightarrow \neg(p\wedge q)\vee(p\wedge q) \end{align*}

where the last term is a tautology.

Hopefully this helps !

answered Jan 16 '22 at 23:00

Átila Correia

15,530

Thank you for this! Can you explain what the symbol ¬ means? – Richard McCormick Jan 16 '22 at 23:01
1

@RichardMcCormick It means the negation operator. – Átila Correia Jan 16 '22 at 23:02
Oh I see, thank you! – Richard McCormick Jan 16 '22 at 23:02
1

@RichardMcCormick You are welcome ! I am glad I could help. – Átila Correia Jan 16 '22 at 23:03

Prove that the proposition $p\to(q\to(p\wedge q))$ is a tautology by using identities.

1 Answers1