## Equivalent propositions

Decide which of the following propositions are equivalent.

$$a \Rightarrow b$$; \quad $$b \Rightarrow a$$; \quad $$a \land b$$;\quad $$\neg a \lor b$$; \quad $$a \Leftrightarrow b$$; \quad $$\neg ( b \Rightarrow \neg a)$$; \quad $$\neg b \Rightarrow \neg a$$; \quad $$\neg (a \land \neg b)$$.

• #### Resolution

We use a truth table:

\begin{array}{cc|cccccccc} a & b & a \Rightarrow b & b \Rightarrow a & a \land b & \neg a \lor b & a \Leftrightarrow b & \neg ( b \Rightarrow \neg a) & \neg b \Rightarrow \neg a & \neg (a \land \neg b)\\ 0 & 0 & 1 & 1 & 0 & 1 & 1 & 0 & 1 & 1 \\ 0 & 1 & 1 & 0 & 0 & 1 & 0 & 0 & 1 & 1 \\ 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ \end{array}
• #### Result

We can divide the propositions into four equivalence groups as follows (by rows):
$$a \Rightarrow b$$;   $$\neg b \Rightarrow \neg a$$;   $$\neg a \lor b$$;   $$\neg (a \land \neg b)$$,
$$b \Rightarrow a$$,
$$a \land b$$;  $$\neg ( b \Rightarrow \neg a)$$,  a
$$a \Leftrightarrow b$$;