Question

Show that$\mathbf{p}\mathbf{|}\left(q|r\right)$ and $\left(p|q\right)\mathbf{|}\mathbf{r}$are not equivalent, so that the logical operator$|$is not associative.

Short answer

For showing, $\mathbf{p}\mathbf{|}\mathbf{\left(}\mathbf{q}\mathbf{|}\mathbf{r}\mathbf{\right)}$and $\mathbf{\left(}\mathbf{p}\mathbf{|}\mathbf{q}\mathbf{\right)}\mathbf{|}\mathbf{r}$are not logically equivalent, prepare the truth table for every statement and also take different values ofpandqsuch as every possible combination ofTand F to check the result for their verification.

Step-by-step solution

Definition of logically equivalent propositions  

Any two propositions are said to belogically equivalentif they have same truth value for any combination of truth values of pand q .

To show that p|(q|r) and p|q|r are not logically equivalent.

Take two variables as inputs such asp,q andr.

Prepare the truth table for $\mathbf{p}\mathbf{|}\left(q|r\right)$and $\left(\mathbf{p}\mathbf{|}\mathbf{q}\right)|\mathbf{r}$.

p	q	r	$\left(\mathrm{q}|\mathrm{r}\right)$	$\mathbf{p}\mathbf{|}\mathbf{\left(}\mathbf{q}\mathbf{|}\mathbf{r}\mathbf{\right)}$	$\left(\mathrm{p}|\mathrm{q}\right)$	$\left(\mathbf{p}\mathbf{|}\mathbf{q}\right)|\mathbf{r}$
T	T	T	F	T	F	T
T	T	F	T	F	F	T
T	F	T	T	F	T	F
T	F	F	T	F	T	T
F	T	T	F	T	T	F
F	T	F	T	T	T	T
F	F	T	T	T	T	F
F	F	F	T	T	T	T

Truth Table

Check the outcomes of $\mathbf{p}\mathbf{|}\mathbf{\left(}\mathbf{q}\mathbf{|}\mathbf{r}\mathbf{\right)}$and $\mathbf{p}\mathbf{|}\mathbf{\left(}\mathbf{q}\mathbf{|}\mathbf{r}\mathbf{\right)}$.

From the truth tables, it is observed that $\mathbf{p}\mathbf{|}\mathbf{\left(}\mathbf{q}\mathbf{|}\mathbf{r}\mathbf{\right)}$and $\mathbf{p}\mathbf{|}\mathbf{\left(}\mathbf{q}\mathbf{|}\mathbf{r}\mathbf{\right)}$are not equivalent as their outcomes are different.

Therefore, the proposition $\mathbf{p}\mathbf{|}\mathbf{\left(}\mathbf{q}\mathbf{|}\mathbf{r}\mathbf{\right)}$ is logically equivalent to$\mathbf{p}\mathbf{|}\mathbf{\left(}\mathbf{q}\mathbf{|}\mathbf{r}\mathbf{\right)}$.