Question

A= {{a}, a}

p(A) =

State the cardinality of set A &

cardinality of p(A) . Please explain in great detail to provide a better understanding .

Answer 1

Few important concepts before I provide the answer to the actual question:
For any set, the cardinality is the number of elements that the set has.
for example : set S = {1, 2, 3}, this set S has 3 elements, so cardinality of set S is 3.

A power set of a set S is set that contain all the subsets of the set S.
Power set of S = P(S) = { { }, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3} }
Notice that empty set { } is also included in P(S), since empty set is also subset of S.
The P(S) contains 8 elements, therefore cardinality of P(S) = 8
The formula for cardinality of |P(S)| = 2ⁿ where n is the cardinality of set S,
In this case n = 3 since |S| = 3, Hence |P(S)| = 2³ = 8
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A= {{a}, a}

Here A contains 2 elements {a} and a, hence cardinality of A is 2,
|A| = 2

Powerset of A contains all subsets of A,
P(A) = {{ }, {{a}}, {a}, {{a}, a} }
Power set has 4 elements and the cardinality of P(A) = 4, |P(A)| = 4

We can also verify the cardinality using the formula: Since |A| = 2, therefore n = 2
|P(A)| = 2ⁿ = 2² = 4, Hence we got the same cardinaliy using the formula.
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