Question

Show that$\mathbf{p}\mathbf{\leftrightarrow }\mathbf{q}$ and $\mathbf{(}\mathbf{p}\mathbf{\to }\mathbf{q}\mathbf{)}\mathbf{\wedge }\mathbf{(}\mathbf{q}\mathbf{\to }\mathbf{p}\mathbf{)}$are logically equivalent

Short answer

It is shown that $\mathbf{p}\mathbf{\leftrightarrow }\mathbf{q}$and $\mathbf{(}\mathbf{p}\mathbf{\to }\mathbf{q}\mathbf{)}\mathbf{\wedge }\mathbf{(}\mathbf{q}\mathbf{\to }\mathbf{p}\mathbf{)}$are logically equivalent.

Step-by-step solution

Definition of Logical equivalence

Logical equivalence is a relationship between two statements in propositional logic.

The two statements are logically equivalent

We write the given expression,

$\mathbf{p}\mathbf{\leftrightarrow }\mathbf{q}$ means pif and only if q

Thus we can write as if pthen qand if then qthen p .

Hence we write it using logical connectives,

$\mathbf{(}\mathbf{p}\mathbf{\to }\mathbf{q}\mathbf{)}\mathbf{\wedge }\mathbf{(}\mathbf{q}\mathbf{\to }\mathbf{p}\mathbf{)}$

Hence, $\mathbf{p}\mathbf{\leftrightarrow }\mathbf{q}$and $\mathbf{(}\mathbf{p}\mathbf{\to }\mathbf{q}\mathbf{)}\mathbf{\wedge }\mathbf{(}\mathbf{q}\mathbf{\to }\mathbf{p}\mathbf{)}$are logically equivalent.