Consider three sets X, Y and Z having 6, 5 and 4 elements respectively. All these 15 elements are distinct. Let S = (X - Y) ∪ Z. How many proper subsets does S have?

Consider three sets X, Y and Z having 6, 5 and 4 elements respectively. All these 15 elements are distinct. Let S = (X - Y) ∪ Z. How many proper subsets does S have? Correct Answer 1023

Given:

Number of elements in X = 6

Number of eleelementsent in Y = 5

Number of elements in Z = 4

All elements are different.

Concept:

If A and B are two sets, then their difference is given by A - B or B - A. 

If A = {2, 3, 4} and B = {4, 5, 6} 

A - B means elements of A that are not the elements of B i.e,

in the above example A - B = {2, 3} 

In general, B - A = {x : x ∈ B, and x ∉ A} 

Proper subset = 2ⁿ - 1,

n is the number of elements

Calculation:

Since, X, Y,  have distnict elements. 

Therefore,

n(X - Y) = 6, n(Z) = 4

⇒ S = (X - Y) ∪ Z

⇒ S = 6 + 4 = 10

Hence,

Number of proper subsets 

N = 2¹⁰ - 1

⇒ N = 1024 - 1

⇒ N = 1023

S have 1023 proper subsets.