For each pair of statements, determine whether the first implies the second. (a)

Question

For each pair of statements, determine whether the first implies the second.

(a) “If I buy a new book, I will be happy”; and “If I do not buy a new book, I will be unhappy.”

(b) “I will be cold if I do not wear a jacket”; and “I will not be cold if I do not wear a jacket.”

(c) “If you smile a lot, your mouth will hurt”; and “If your mouth hurts, you will smile a lot.”

(d) “A warm house implies a warm bathroom”; and “A cold bathroom implies a cold house.”

(e) “Eating corn implies that I will have to floss my teeth”; and “Not having to floss my teeth implies that I will eat corn.”

(f) “Going to the beach is sufficient for me to have fun”; and “Not going to the beach is sufficient for me not to have fun.”

Explanation / Answer

(a) is wrong, since, buying a new book ensures happiness, but that does not imply that, not buying a book makes unhappy.

(b) is wrong

(c) is wrong

(d) is correct, i.e. a warm house implies a warm bathroom.

Contrapositively, not a warm bathroom implies not a warm house

i.e. cold bathroom implies cold house

(e) is wrong

I.e. eating corn implies flossing my teeth.

Contrapositively, not flossing my teeth would imply I did not have corn.

(f) is wrong

Going to the beach implies having fun

Contrapositively , not having fun implies I did not go to the beach

Hence in all the cases above, only in (d) the first statement implies the second. For all the others, it is not true.

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