Chapter 1 - Section 1.3 - Propositional Equivalences - Exercises - Page 34: 4

$a.$ $\left[\begin{array}{lll} p & q & r\\ \hline & & \\ T & T & T\\ T & T & F\\ T & F & T\\ T & F & F\\ F & T & T\\ F & T & F\\ F & F & T\\ F & F & F \end{array}\right.$ $\begin{array}{ll} p\vee q & (p\vee q)\vee r\\ \hline & \\ T & T\\ T & T\\ T & T\\ T & T\\ T & T\\ T & T\\ F & T\\ F & F \end{array}$ $\left.\begin{array}{ll} q\vee r & p\vee(q\vee r)\\ \hline & \\ T & T\\ T & T\\ T & T\\ F & T\\ T & T\\ T & T\\ T & T\\ F & F \end{array}\right]$ $b.$ $\left[\begin{array}{lll} p & q & r\\ \hline & & \\ T & T & T\\ T & T & F\\ T & F & T\\ T & F & F\\ F & T & T\\ F & T & F\\ F & F & T\\ F & F & F \end{array}\right.$ $\begin{array}{ll} p\wedge q & (p\wedge q)\wedge r\\ \hline & \\ T & T\\ T & F\\ F & F\\ F & F\\ F & F\\ F & F\\ F & F\\ F & F \end{array}$ $\left.\begin{array}{ll} q\wedge r & p\wedge(q\wedge r)\\ \hline & \\ T & T\\ F & F\\ F & F\\ F & F\\ T & F\\ F & F\\ F & F\\ F & F \end{array}\right]$

$a.$ The columns for $(p\vee q)\vee r$ and $p\vee(q\vee r)$ are identical. $b.$ The columns for $(p\wedge q)\wedge r$ and $p\wedge(q\wedge r)$ are identical.