Please help thank you Let S and T be two infinite sets of n

Please help, thank you :)

Let S and T be two infinite sets of numbers.. (a) What does the statement \"S and T have the same cardinality\" mean? (b) Suppose S notequalto T (i.e. S is a proper subset of T). Is it possible for S and T to have the same cardinality? Justify your answer.

Solution

(a)

It means one can define a bijection from S to T.

(b)

Yes it is possible

Consider the set of all integers , Z and consider the set of all even integers, S

we can define the map,h, from Z to S:h(x)=2x

This is a one-to-one and onto map. Hence a bijection.

Hence, Z and S have the same cardinality but S is a proper subset of Z.

Note that this is possible only for infinite sets.