Let 𝑈 = {1,2, … , 𝑛}. Let 𝐴 = {(𝑥, 𝑋)|𝑥 ∈ 𝑋, 𝑋 ⊆ 𝑈}. Consider the following two statements on |𝐴|

August 2, 2023

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Q.	Let 𝑈 = {1,2, … , 𝑛}. Let 𝐴 = {(𝑥, 𝑋)\|𝑥 ∈ 𝑋, 𝑋 ⊆ 𝑈}. Consider the following two statements on \|𝐴\|. I. \|𝐴\| = 𝑛2^𝑛−1 II. \|𝐴\| = ∑^𝑛𝑘=1 𝑘(^𝑛) Which of the above statements is/are TRUE?
	(A) Only I	(B) Only II
	(C) Both I and II	(D) Neither I nor II

Ans: Both I and II

Given, A = {(x, X)∣ x∈X, X⊆U }, where U = {1, 2, …,n}.

As we know that The number of k element subsets of a set U with n elements = ⁿC_k.

The number of possible ordered pairs (x, X) where x ∈ X is k⋅ ⁿC_k for a given value of k from 1 to n. So total number of ordered pairs in A,

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