MathCS.org - Real Analysis: Examples 2.2.4(a):

Examples 2.2.4(a):

If S = {1,2,3}, then what is P(S) ? What is the power set of the set S = {1, 2, 3, 4} ? How many elements does the power set of S = {1, 2, 3, 4, 5, 6} have ?

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The power set is the set of all subsets of a given set.

For the set S = {1,2,3} this means:

subsets with 0 elements: 0 (the empty set)
subsets with 1 element: {1}, {2}, {3}
subsets with 2 elements: {1,2}, {1,3}, {2,3}
subsets with 3 elements: S

Hence:

P(S) = {0, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, S}

Therefore, we have:

card(S) = 3 and card(P(S)) = 8 = 2³

For the set S = {1,2,3,4} this means:

subsets with 0 elements: 0 (the empty set)
subsets with 1 element: {1}, {2}, {3}, {4}
subsets with 2 elements: {1,2}, {1,3}, {1,4}, {2,3}, {2,4}, {3,4}
subsets with 3 elements: {1,2,3}, {1,2,4}, {1,3,4} {2,3,4}
subsets with 4 elements: S

Therefore, we have:

card(S) = 4 and card(P(S)) = 16 = 2⁴

Finally, if S = {1,2,3,4,5,6} then, based on the above examples, we would suspect that

card(S) = 6, therefore card(P(S)) = 2⁶ = 64

In fact, if a set S contains n elements, then its power set will contain 2ⁿ elements. This can be proved by induction as an exercise.

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Examples 2.2.4(a):