Given a set A with n elements, we define the power set of A to be the set contai

Question

Given a set A with n elements, we define the power set of A to be the set containing every subset of A. For example, the power set of {a, b, c} is {{}, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}} Note that {a, b, c} contains 3 elements, and the power set of {a, b, c} contains 2^3 elements. Recall that the set of bit-strings with 3 bits also contains 2^3 elements: {000,001, 010, 011, 100, 101, 110, 111} For this problem, find a bijection between the power set of A and the set of bit strings with n bits. (On a side note, this helps to prove that the power set of A has 2^n elements)

Explanation / Answer

When there are n elements in set A,

use n bits, in which each bit will represent presence of one element.

Suppose there are three elements in set A = {a,b,c} .

here if three bits are used then first bit will represent a, second bit will represent b and third bit will represent c.

There are such 23 combinations possible.

So there can be 23 subsets of set A.

along with 000 which shows empty set { } and 111 which shows set A itself {a,b,c}.

Hence each subset can be mapped with one such combination.

Hence, each element in powerset will be mapped to one string with n bits.

and each n bits combination represents one subset of set A which is element of power set of set A.

Thus, there is a bijection between set of binary strings with length n and powerset of set A with element n.

both sets will have 2n elements.

if you have any doubts, you can ask in comment section.

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