## Associativity of implication

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.