Associativity of implication
Task number: 2780
Are the propositions \((a\Rightarrow b)\Rightarrow c\ \) and \(\ a\Rightarrow (b\Rightarrow c)\) equivalent?
Hint
Try replacing \(a \Rightarrow b\) with \(\neg a \lor b\).
Resolution
\(\Phi=(a\Rightarrow b)\Rightarrow c \iff \neg(\neg a \lor b)\lor c \iff (a \land \neg b) \lor c\)
\(\Psi=a\Rightarrow (b\Rightarrow c) \iff \neg a \lor \neg b \lor c \)
So \(\Psi\) is false only when \(a=1, b=1, c=0\), while \(\Phi\) is false when \(a=1, b=1, c=0\); also when \(a=0, b=1, c=0\); also when \(a=0, b=0, c=0\).
Altenatively we can use a truth table and reach the same conclusion:
\begin{array}{ccc|cccc} a & b & c & a \Rightarrow b & (a\Rightarrow b)\Rightarrow c & b \Rightarrow c & a\Rightarrow (b\Rightarrow c)\\ 0 & 0 & 0 & 1 & 0 & 1 & 1 \\ 0 & 0 & 1 & 1 & 1 & 1 & 1 \\ 0 & 1 & 0 & 1 & 0 & 0 & 1 \\ 0 & 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 0 & 0 & 0 & 1 & 1 & 1 \\ 1 & 0 & 1 & 0 & 1 & 1 & 1 \\ 1 & 1 & 0 & 1 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ \end{array}Result
The propsitions are not equivalent.
