Q.Using truth tables, prove the logical equivalence $p \rightarrow q \equiv \sim p \vee q$.

Build one table with columns for $p \rightarrow q$ and for $\sim p \vee q$, then compare them row by row.

Construct a single truth table carrying both patterns.

Compare the two relevant columns. The column for $p \rightarrow q$ is $T, F, T, T$. The column for $\sim p \vee q$ is: row 1 $F \vee T = T$; row 2 $F \vee F = F$; row 3 $T \vee T = T$; row 4 $T \vee F = T$, i.e. $T, F, T, T$.

The two columns are identical in every row, so by definition the patterns are logically equivalent: $p \rightarrow q \equiv \sim p \vee q.$ …

The mistake here is negating the wrong part - writing p or ~q instead of ~p or q. In p -> q = ~p or q, it is the a …