Question

If n ( A ) = 6 , then number of elements in its Power set are

Answer 1

Answer:

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Step-by-step explanation:

ANSWER

If a non-empty set A contains n elements, then its power set contains 2

n

elements.

This can be proved using mathematical induction.

Base Case: suppose ∣A∣=0⟹A=ϕ. But, empty set is only subset of itself. So, ∣P(A)∣=1=2

0

.

Now, suppose ∣A∣=n.

By induction hypothesis, we know that ∣P(A)∣=2

n

⟶1

Let B be a set with (n+1) elements, B=A∪{a}

Now, there are 2 kinds of subsets of B: those that include

′

a

′

and those that don't.

The first ones are exactly the subsets of X which do not contain

′

a

′

and there are 2

n

of them.

The second one are of the form C∪{a}, where C∈P(A). since there are 2

n

possible choices for C, there must be exactly 2

n

subsets of B of which

′

a

′

is an element.

∴∣P(B)∣=2

n

+2

n

=2

n+1

.

so, if set has n elements, then power set has 2

n

elements.

Hence proved.

Answer 2

Answer:

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