Question

Verify whether the following statements is true or false.
Give a reason to support your answer.
(i) A set can have infinitely many subsets.

Answer 1

Hint:Take the subsets of an infinite set as A. Now find its power set and by using cantor's theorem, find the cause of infinite sets and finite sets.

Complete step-by-step answer:
For an infinite set has more subsets than its own infinite number of elements. The sets of subsets of a set A is called its power set and it is represented as P (A). An infinite set can definitely have infinitely many subsets, as there are infinite objects in that set, which are alone a subset by themselves.
The number of subsets in this infinite set are infinitely greater than the number of objects in the set itself.
There are different types of infinities. These infinites have a ranking just like the natural number. We can call the set of real numbers, which is infinitely bigger than A1 is A2. Thus the list goes on.
Thus according to Cantor’s theorem is a fundamental result which states that for any set A, the set of all subsets of A has a strictly greater cordiality than A itself. For finite sets Cantor’s theorem can be seen to be true by simple enumeration of subsets.
Counting the empty set as a subset, a set with n members has a total of $2^n$ subsets, so that if n (A) = n i.e number of elements present in set A, then P (A) =\[{{2}^{n}}\] i.e number of subsets of set A containing n elements , and the theorem holds because \[{{2}^{n}}>n\] for all non – negative integer.If value of n is infinity i.e number of elements of set A is infinite then number of subsets will be greater than the number of elements in set itself For Ex:Set of natural numbers.
Hence we can say that A set can have infinitely many subsets on the given statement is true.

Note: For a set, \[A=\left\{ 1,2 \right\},\]it will have finite number of subsets such as, \[B=\left\{ 1, \right\},C=\left\{ 2 \right\},D=\left\{ 1,2 \right\},E\ne 0\] will be only subsets of A.Students should remember Cantor’s theorem which states that for any set A, the set of all subsets of A has a strictly greater cordiality than A itself.