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

on |A|.

I. |A| = n2n - 1

II. \(\left| A \right| = \mathop \sum \limits_{k = 1}^n k\left( {\begin{array}{*{20}{c}} n\\ k \end{array}} \right)\)

Which of the above statements is/are TRUE?
1. Only I
2. Only II
3. Both I and II 
4. Neither I nor II

1 Answer

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Best answer
Correct Answer - Option 3 : Both I and II 

Let U = {1, 2}

All Possible subsets of U = {ϕ, {1}, {2}, {1, 2}}

A = (x, X), x ∈ X and X ⊆ U

x can be only {ϕ, 1, 2}

When x = 1

X = (1, {1})

X = {1, {1, 2}}

When x = 2

X = {2, {2}}

X = {2, {1, 2}}

Therefore, total elements in A, |A| = 2 + 2 = 4.

Option 1:

|A| = n × 2n - 1 = 2 × 22 - 1 = 4

Option 2:

\(\left| A \right| = \mathop \sum \limits_{k = 1}^n k\left( {\begin{array}{*{20}{c}} n\\ k \end{array}} \right) = 1 \times \left( {\begin{array}{*{20}{c}} 2\\ 1 \end{array}} \right) + 2 \times \left( {\begin{array}{*{20}{c}} 2\\ 2 \end{array}} \right)\)

\(\left| A \right| = 2 + 2 = 4\)

Both the options are correct.

Important Points:

x = ϕ and X = ϕ is not considered since ϕ ∈ ϕ is not true.

Although we cannot generalize just from one example but in general both the cases always hold true for given conditions

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