Answer to Question #105391 in Discrete Mathematics for ea

Question

A is sufficient for B’ is equivalent to ‘the negative of A is necessary for the
negative of B’. is it true or false?

Answer 1

A is sufficient for B can be written as $A \to B$ ---(1)

The negative of A is necessary for the negative of B can be written as $\neg B \to \neg A$ ---(2)$(2)\implies \neg B \to \neg A \iff \neg (\neg B) \lor \neg A$

$\iff \neg (\neg B \land A)$ (Using DeMorgan's laws)

$\iff \neg(\neg(\neg (\neg B \land A)))$ (As $A \iff \neg\neg A$ )

$\iff \neg(\neg B\land A)$ (As $\neg\neg A \iff A$ )

$\iff B \lor \neg A\iff \neg A \lor B$ (Using DeMorgan's laws)

$\iff A \to B \iff (1)$

Thus, (2) is equivalent to (1).

Hence, the statement 'A is sufficient for B’ is equivalent to ‘the negative of A is necessary for the

negative of B’ is TRUE.