Set |Subset | Volume 1 Q6

Question

Question

If A and B are subsets of set X = { 1,2,3,  ...  99,100 } and A # B denote the set of all elements of X which belogs to exactly A or B . Total number of subsets of X such that A#B = { 2,4,6,.. 98,100} is _____

My approach

Every even Number can appear in either of 2 subsets (A-B) or (B-A) ie  Every even number can appear either in only A or in only B

Every odd number can appear only in A$\cap$ B

how to proceed further ...?

Comments

There are two types of elements in X.

1) Odd Elements : 

There are total 50 odd elements in X.

Each of these 50 odd elements will have 2 choices to get themselves excluded from the set A#B:

    Either be in both the subsets A & B

    Or be in none of the subsets A & B.

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2) Even Elements:

There are total 50 even elements in X.

Each of these 50 elements will have two choices in order to get themselves included in the set A#B:

    Either be in A only and not in B.

    Or be in B only and not in A. 

Thus irrespective of being odd or even, every element of the set X has got two choices, and since X has 100 elements in total, we have got 2¹⁰⁰  ways to choose the set A#B.

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Thanks :)

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nice xplanation anurag..

Answer 1

But in question why they mention total no of subset of x such that a exor b={2,3,....100}

Why not they start from 1