Equivalent propositions
Task number: 2778
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
\(a \Leftrightarrow b\);
