TIFR CSE 2019 | Part A | Question: 1

Question

Let $X$ be a set with $n$ elements. How many subsets of $X$ have odd cardinality?

• A. $n$                                        
• B. $2^n$
• C. $2^{n/2}$
• D. $2^{n-1}$
• E. Can not be determined without knowing whether $n$ is odd or even

Answer 1

For n=0 then clearly there is 1 such subset, the empty set.

If n>0 list the elements of X as

                                                           x1,x2,x3, x4 …..........,xn

A subset S⊆X with an even number of elements is determined by its intersection with {x1,…,xn−1}: if the intersection has an even number of elements then xn∉S, and if it has an odd number of elements then xn∈S

.

Thus the number of subsets of X with an even number elements is equal to the number of subsets of {x1,…,xn−1}, which is 2^(n−1)

Answer 2

(D) is correct answer .because of

total sub group=2^n

half of even and half of odd .