Question

Let X = {1, 2, 3, 4, 5}.The number of different ordered pairs (Y , Z) that can formed such that Y ⊆ X, Z ⊆ X and y ⋂  Z is empty, is

(a) 5² 

(b) 3⁵ 

(c) 2⁵ 

(d) 5³

Comments

Thanks so much bro ...u answered it in perfect way ....other sites just telling with permutations which does not show any logic in it

Answer 1

Correct option (b) 3⁵ 

Explanation:

Let us explain anyone of the above 6 rows say third row. In third row ,

Number of elements in Y = 2

Number of ways to select Y = ⁵C₂ ways

Because any 2 elements of X can be part of y.

Now, if Y contains any 2 elements, then these 2 elements cannot be used in any way to construct Z because we want Y U Z = ϕ. And from the remaining 3 elements which are not present in Y, 2³ subsets can be made each of which can be equal lo Z and still Y ⋂ Z = ϕ will be true.  Hence, total number of ways to construct sets Y and Z such that Y ⋂ Z = ϕ