Chapter 1.2, Problem 5E

Using the properties in the text together with the absorption properties given in Exercise 4, where needed, establish each of the following logical equivalences.

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

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

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

(d) $\neg \left[\left(p\leftrightarrow q\right)\vee \left(p\wedge \left(\neg q\right)\right)\right]\iff \left[\left(p\leftrightarrow \left(\neg q\right)\right)\wedge \left(\left(\neg p\right)\vee q\right)\right]$

(e) $\left[\left(p\wedge \left(\neg q\right)\right)\wedge \left(\left(p\wedge \left(\neg q\right)\right)\vee \left(q\wedge \left(\neg r\right)\right)\right)\right]\iff \left[p\wedge \left(\neg q\right)\right]$

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

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