Question

A set contains $n$ elements. The power set of this set contains
(a) ${{n}^{2}}$ elements
(b) ${{2}^{\dfrac{\lambda }{2}}}$ elements
(c) ${{2}^{n}}$ elements
(d) $n$ elements

Answer 1

We will look at the definition of a set and its cardinality. Then we will see the definition of a power set. We will count the elements in the power set to obtain the cardinality of the power set. Given the cardinality of a set, We will obtain an expression that gives us the number of elements in the power set. We will use the multiplication principle of counting in this.

Complete step by step answer:
A set is defined as a collection of well-defined, distinct objects. The cardinality of a set is the number of elements in the set. We have a set which contains $n$ elements. Let this set be set $A$. Now, we will look at the definition of a power set. A power set is defined as a set of all possible subsets of a set. Next, we have to find the number of elements in the power set of set $A$. This means that we have to count the number of all possible subsets of set $A$.
We can see that, in any subset, there are two choices for an element of set $A$; either the element belongs to the subset or it does not belong to the subset. Since we have $n$ elements, the choices using the multiplication principle of counting, will be $\underbrace{2\times 2\times \ldots \times 2}_{n\text{ times}}={{2}^{n}}$. Hence, the cardinality of the power set is ${{2}^{n}}$.

So, the correct answer is “Option C”.

Note: It is not possible to explicitly count every element in the power set if the cardinality of the given set is countably infinite or infinite. The multiplication principle of counting states that if there are $p$ ways of doing a thing and there are $q$ ways of doing another thing, then there are $p\times q$ ways of doing both the things together.