Construct the truth table?

Question

Any body help me .. How to solve this?

(i) $(p\land q)\to (p \leftrightarrow (q \lor r))$

(ii) $(p \leftrightarrow q) \leftrightarrow ((p\land q) \lor (\neg q \land \neg p))$

(iii) $(p \lor q) \leftrightarrow \neg(\neg p \land \neg q)$

Answer 1

Perhaps this will be helpful: i.
$((p\wedge q)\to(p\equiv(q\vee r)))$
$a:p\wedge q$
$b:q\vee r$
$c:p\equiv b$
$d:a\to c$
$\begin{matrix} p&q&r&a&b&c&d\\ 0&0&0&0&0&1&1\\ 0&0&1&0&1&0&1\\ 0&1&0&0&1&0&1\\ 0&1&1&0&1&0&1\\ 1&0&0&0&0&0&1\\ 1&0&1&0&1&1&1\\ 1&1&0&1&1&1&1\\ 1&1&1&1&1&1&1\\ \end{matrix}$

Answer 2

Following Mauro's suggestion, I'll give you a general algorithm using truth-tables, focusing on (i).

Let n be the number of distinct propositional letters ocurring in the formula (e.g. for (i) you'll have |{p, q, r}| = 3). Generate a truth-table of 2ⁿ rows. In the case of (i), for example, n = 2³ = 8. In order to prove (i), consider those rows out of the 8 that make (p ∧ q) true. Now check, do those rows make (p ↔ (q ∨ r)) true? To answer those two questions, you need to look at the truth-conditions for the connectives ∧, →, ↔, ∨. Each row of the truth-table assigns a value (⊤ or ⊥) to p, q, and r. The truth-conditions allow you to evaluate the truth-value of compound expressions like (p ∧ q) using nothing more than the truth-values of p and q.

Once you know how to calculate the truth-value of compounds using the truth-values of the atoms, you can answer all three of your questions. When you do, show your work if you need guidance.