**Determine whether each of these sets is the power set of a set. where a and b are distinct elements. a) ∅ b) {∅.{∅}} c) {∅, {a}. {∅, a)} d) {∅, {a}. {b}. {a, b}}**

The **Answer to the Question**

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**Here's the Solution to this Question**

a) The aim is to determine whether the set ∅ is a power set of a set or not.

By definition, given a set S, the power set of S is a set of all subsets of the set S. The power set of S is denoted by P(S).

The power set of every set includes at least the empty set, so the power set cannot be empty.

Thus we may conclude that the given set is not a power set of any set.

b) The objective is to determine whether the set {∅.{∅}} is a power set of a set or not.

Given a set S, the power set of S is a set of all subsets of the set S. The power set of S is denoted by

P(S).

Using the result, for a set A if $\mid$ A $\mid$ = k, then the cardinality of P(A) = 2k

The number of elements in A = { a } is 1, so the cardinality of P(A) = 21 = 2.

The power set of { a } is {∅.{∅}}.

Thus, the given set is the power set of the set A = { a }

c) The aim is to determine whether the set {∅, {a}. {∅, a)} is a power set of a set or not.

The number of elements in A ={ a } is 1, so the cardinality of P(A) = 21 = 2.

But the set {∅, {a}. {∅, a)} contains three elements, so this cannot be the power set of A={ a }

Therefore, the given set is not a power set of any set.

d) The objective is to determine whether the set {∅, {a}. {b}. {a, b}} is a power set of a set or not, where the elements a and b are distinct.

The number of elements in A = { a, b } is 2, so the cardinality of P(A) = 22 = 4

The power set of {a, b} is {∅, {a}. {b}. {a, b}} .

Thus, the given set is the power set of A = {a, b}