Given: p is true Prove: p → q is true Assume ~q is true. Then ~q → r, and r → s. Since s → ~p, ~q → ~p by the law of syllogism. Therefore, p → q is true. What type of proof is illustrated above? A. proof by contradiction B. proof by contraposition C. proof by law of detachment D. proof by law of syllogism

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Mathematics

Acevedo

31 July, 11:43

Given: p is true Prove: p → q is true Assume ~q is true. Then ~q → r, and r → s. Since s → ~p, ~q → ~p by the law of syllogism. Therefore, p → q is true. What type of proof is illustrated above? A. proof by contradiction B. proof by contraposition C. proof by law of detachment D. proof by law of syllogism

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Koen Herring · Answer 1

Given the following proof:

p → q is true Assume ~q is true. Then ~q → r, and r → s. Since s → ~p, ~q → ~p by the law of syllogism. Therefore, p → q is true.

We can see that the conclusion was drawn from the fact that since ~q → ~p, then p → q.

This is known as contraposition.

Contraposition in logic is the conversion of a proposition from, for example: all A is B to all not-B is not-A.