Chapter 1.2, Problem 1E

To determine

To prove: The following properties of logical equivalence.

Idempotence:

(i) $(p\vee p)\iff p$

(ii) $(p\wedge p)\iff p$

Commutativity:

(i) $\left(p\vee q\right)\iff \left(q\vee p\right)$

(ii) $\left(p\wedge q\right)\iff \left(q\wedge p\right)$

Associativity:

(i) $\left(\left(p\vee q\right)\vee r\right)\iff \left(p\vee \left(q\vee r\right)\right)$

(ii) $\left(\left(p\wedge q\right)\wedge r\right)\iff \left(p\wedge \left(q\wedge r\right)\right)$

Distributivity:

(i) $\left(p\vee \left(q\wedge r\right)\right)\iff \left(\left(p\vee q\right)\wedge \left(p\vee r\right)\right)$

(ii) $\left(p\wedge \left(q\vee r\right)\right)\iff \left(\left(p\wedge q\right)\vee \left(p\wedge r\right)\right)$

Double Negation:

$\neg \left(\neg p\right)\iff p$

De Morgan’s Laws:

(i) $\neg \left(p\vee q\right)\iff \left(\left(\neg p\right)\wedge \left(\neg q\right)\right)$

(ii) $\neg \left(p\wedge q\right)\iff \left(\left(\neg p\right)\vee \left(\neg q\right)\right)$

(i) $\left(p\vee 1\right)\iff 1$

(ii) $\left(p\wedge 1\right)\iff p$

(i) $\left(p\vee 0\right)\iff p$

(ii) $\left(p\wedge 0\right)\iff 0$

(i) $\left(p\vee \left(\neg p\right)\right)\iff 1$

(ii) $\left(p\wedge \left(\neg p\right)\right)\iff 0$

(i) $\neg 1\iff 0$

(ii) $\neg 0\iff 1$

$\left(p\to q\right)\iff \left[\left(\neg q\right)\to \left(\neg p\right)\right]$

$\left(p\to q\right)\iff \left[\left(\neg p\right)\vee q\right]$

$\left(p\leftrightarrow q\right)\iff \left[\left(p\to q\right)\wedge \left(q\to p\right)\right]$