Answer to Question #105391 in Discrete Mathematics for ea

Question #105391
A is sufficient for B’ is equivalent to ‘the negative of A is necessary for the
negative of B’. is it true or false?
1
Expert's answer
2020-03-16T13:42:32-0400

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

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

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

    ¬(¬(¬(¬BA)))\iff \neg(\neg(\neg (\neg B \land A))) (As A    ¬¬AA \iff \neg\neg A )

    ¬(¬BA)\iff \neg(\neg B\land A) (As ¬¬A    A\neg\neg A \iff A )

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

    AB    (1)\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.


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