12) Let S1 be a countable set, S2 a set that is not countable, and S1...
12) Let S1 be a countable set, S2 a set that is not countable, and S1 ⊂ S2.
a) Show that S2 must then contain an infinite number of elements that are not in S1.
b) Show that in fact S2 − S1cannot be countable.
Homework Answers
a) S1 is a countable set implies that there exists a bijection from a subset of the natural numbers to S1 and S2 is not countable implies that no bijection exists from any subset of N to S2. Let S3 be the set of elements in S2 that are not in S1. Let S3 be finite. Then S3 and S1 both are countable which implies their union S2 should also be countable. But that is a contradiction. Hence S3 cannot be finite.
b) Let us assume S3 is countable. Since S1 and S3 are both countable, their union S2 should also be countable, which leads to a contradiction. Thus S3, which is the same as S2-S1 cannot be countable.
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