Question

p ↔ q ≡ (p ∧ q) ∨ (∼ p ∧ ∼ q)

Without Using Truth Table, Prove It.

Answer 1

$\textbf{To prove:}$

$\bf\,p{\iff}q\;\equiv\;(p{\wedge}q)\vee({\neg}p{\wedge}{\neg}q)$

$\textbf{Solution:}$

$\text{First we form the truth tables for both the statements}$

$\text{After that we compare the last columns whether they are identitical}$

$\begin{array}{|c|c|c|}\cline{1-3}\bf{p}&\bf{q}&\bf{p{\iff}q}\\\cline{1-3}T&T&T\\T&F&F\\F&T&F\\F&F&T\\\cline{1-3}\end{arrary}$

$\begin{array}{|c|c|c|c|c|c|c|}\cline{1-7}\bf{p}&\bf{q}&\bf\neg{p}&\bf\neg{q}&\bf{p{\wedge}q}&\neg{p}{\wedge}\neg{q}&\bf(p{\wedge}q)\vee({\neg}p{\wedge}{\neg}q)\\\cline{1-7}T&T&F&F&T&F&T\\T&F&F&T&F&F&F\\F&T&T&F&F&F&F\\F&F&T&T&F&T&T\\\cline{1-7}\end{array}$

$\text{Comparing last columns of both the tables, they are identical}$

$\therefore\bf\,p{\iff}q\;\equiv\;(p{\wedge}q)\vee({\neg}p{\wedge}{\neg}q)$