Suppose S is an infinite set. Assuming that the axiom of choice holds, which of the following is true?

Concept:

A function f : A → B is bijective if |A| = |B| where |A| is the cardinality of A.

Explanation:

S is a infinite set

(1): If S is set of real number i.e., S = \(\mathbb R\) then |S| = C

and cardinality of set of rational number = |\(\mathbb Q\)| = \(\aleph_0\)

Since |S| ≠ |\(\mathbb Q\)| so S is not in bijection with the set of rational numbers.

Option (1) is false

(2): If S is set of integer then |S| = \(\aleph_0\) and |\(\mathbb R\)| = C so |S| ≠ |\(\mathbb R\)|  

So S is not in bijection with the set of real numbers.

Option (2) is false

(4): If S is the set of integers then |S| = \(\aleph_0\)

So cardinality of power set of S = \(2^{\aleph_0}\) = C

Hence S is not in bijection with the power set of S.

Option (4) is false

(3): If S = set of integer then |S| = \(\aleph_0\) and cardinality of S × S = |S × S| = |\(\aleph_0\) × \(\aleph_0\)| = \(\aleph_0\)

Similarly for any infinite set we will be the cardinality of S is same as the cardinality of S × S

Hence S is in bijection with S × S. 

Option (3) is correct