Question

can a set have infinitly many subsets ?explain

Answer 1

Yes, any infinite set actually has strictly more subsets than its own infinite number of elements.

The set of subsets of a set, SS, is called its power set, P(S)P(S).

The number of elements in a set is called its cardinality and is denoted |S||S|.

Since for every x∈S,{x}∈P(S)x∈S,{x}∈P(S), SS being infinite implies P(S)P(S) is also infinite.

If SS is finite with |S|=n|S|=n elements, we can show |P(S)|=2n|P(S)|=2n elements, since every subset can be uniquely defined by whether each element is (independently) a member of the subset. So any finite set has only finitely many subsets, but |P(S)|>|S||P(S)|>|S|.

In fact Cantor’s Theorem proves |P(S)|>|S||P(S)|>|S|for any set SS. If the cardinality of SS is κκ (finite or transfinite), we denote the cardinality of P(S)P(S)as 2κ2κ. So not only does an infinite set have infinitely many subsets, but it has strictly more subsets than it has members. That is the elements of the powerset cannot be put in one-to-one correspondence with the elements of the underlying set.