SOLUTION: Let a, b ∈ R. Prove using contradiction that if a ∈ Q and ab ∉ Q then b ∉ Q.

Question 1042855: Let a, b ∈ R. Prove using contradiction that if a ∈ Q and ab ∉ Q then b ∉ Q.
Answer by ikleyn(52777) (Show Source): You can put this solution on YOUR website!
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Let a, b ∈ R. Prove using contradiction that if a ∈ Q and ab ∉ Q then b ∉ Q.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Let assume that b ∈ Q.   (1)

Then  ab ∈ Q.

But  ab ∉ Q, according to the condition.

Contradiction.

It means that the original assumption (1) is wrong.

Hence,  b ∉ Q.

QED.

Proved.

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(*) Actually, it is usually considered as trivial and obvious statement . . .

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SOLUTION: Let a, b ∈ R. Prove using contradiction that if a ∈ Q and ab ∉ Q then b ∉ Q.