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Can anyone explain the opposite

I am trying to build a contrapositive for the following expression: If A is 0 or B is 0, then A * B is 0.

Here's my attempt: If A * B is not 0, then A is not 0 or B is not 0.

The original statement is correct, but the inconsistency is false since both A and B must be nonzero for A * B to be nonzero ... I am doing something wrong

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3 answers


Yes.

The counterpositive "If P then Q" is "If not Q, then not P".



Thus, the contraceptive "If A is 0 or B is 0, then A *

B is 0" means "If A *

B is not 0, then not (A is 0 or B is 0)",

And “no (A is 0 or B is 0)” means “A is not 0 and B is not 0”, so the contraindicated should be “If A *

B is not 0, then A is not 0 and B is not 0”. What you expect :-)

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you need to change the main "or" to a and. see http://en.wikipedia.org/wiki/De_Morgan%27s_laws



so: If A * B is not 0, then A is not 0 and B is not 0.

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Yes, you did something wrong. NOT (A or B) = NOT (A) and NOT (B). You have neglected to change "or" to "and" while propagating "no". ( De Morgan )

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