Let π = {1,2, β¦ , π}. Let π΄ = {(π₯, π)|π₯ β π, π β π}. Consider theβ¦
2019
Let π = {1,2, β¦ , π}. Let π΄ = {(π₯, π)|π₯ β π, π β π}. Consider the following two statements on |π΄|.
I.Β Β Β Β Β \(\mid A \mid = n2^{n-1}\)
II.Β Β Β Β \(\mid A \mid = \Sigma_{k=1}^{n} k \begin{pmatrix} n \\ k \end{pmatrix}\)
Which of the above statements is/are TRUE?
- A.
Only I
- B.
Only II
- C.
Both I and II
- D.
Neither I nor II
Attempted by 260 students.
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Correct answer: C
Let U = {1, 2}
All Possible subsets of U = {Ο, {1}, {2}, {1, 2}}
A = (x, X), x β X and X β U
x can be only {Ο, 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 Γ 2βΏβ»ΒΉ = 2 Γ 2Β²β»ΒΉ = 4
Option 2:
|A| = Ξ£ k C(n,k)
= 1 Γ C(2,1) + 2 Γ C(2,2)
|A| = 2 + 2 = 4
Both the options are correct.
Important Points:
x = Ο and X = Ο is not considered since Ο β Ο is not true.
Although we cannot generalize just from one example but in general both the cases always hold true for given conditions.