Set Elements Identification

The given set is represented as S = {2, {1, 4}}.

To find the number of elements in the power set P(S), we first need to identify the distinct elements within the set S.

The elements of set S are:

• The number 2.
• The set {1, 4}.

Therefore, the set S contains 2 distinct elements. Let's denote the number of elements in set S as '$n$'. So, we have $n = 2$.

Power Set Concept and Calculation

The power set of a set S, denoted as P(S), is the set containing all possible subsets of S, including the empty set (denoted as {} or $\emptyset$) and the set S itself.

A fundamental property of power sets is that if a set S has '$n$' elements, its power set P(S) will have $2^n$ elements (subsets).

In this case, since the set S = {2, {1, 4}} has $n = 2$ elements, the number of elements in its power set P(S) can be calculated using the formula:

Number of elements in P(S) = $2^n$

Substituting the value of $n=2$:

Number of elements in P(S) = $2^2$ = 4

Listing Subsets of S

To verify, let's list all the subsets of S = {2, {1, 4}}:

• The empty set: {}
• Subsets with one element: {2} and {{1, 4}}
• Subset with two elements: {2, {1, 4}}

The power set P(S) is { {}, {2}, {{1, 4}}, {2, {1, 4}} }. Counting these subsets, we find there are exactly 4 elements in the power set P(S).

Conclusion

The number of elements in the power set P(S) of the set S = {2, {1, 4}} is 4.