GATE | GATE CS 2021 | Set 2 | Question 60

Let S be a set of consisting of 10 elements. The number of tuples of the form (A,B) such that A and B are subsets of S, and A⊆B is ___________ .
(A) 59049
(B) 1024
(C) 50049
(D) 59000


Answer: (A)

Explanation:

Method 1:

Lets take |S|=1

S={1}

A⊆B, and A,B are subsets of S.

A	B
ɸ	ɸ
ɸ	{1}
{1}	{1}

3¹=3

Lets take |S|=2

S={1,2}

P(S)= {ɸ, {1}, {2}, {1,2} }

A	B
ɸ	ɸ
ɸ	{1}
ɸ	{2}
ɸ	{1,2}
{1}	{1}
{2}	{2}
{2}	{1,2}
{1,2}	{1,2}

3²=9

for n=3, 3³=27.

…………………………….

similarly n=4, 3¹⁰=59049.

Method 2:

We can take cases like when |A|=0, |A|=1, |A|=2 …..|A|=n, where is the cardinality of the set S.

|A|= 0. means A=ɸ, so there will be ⁿC₀ x 2ⁿ pairs of A and B.

|A|= 1,  there will be ⁿC₁ x 2^n-1 pairs of A and B.

|A|= 2.   there will be ⁿC₂ x 2^n-2 pairs of A and B.

…………………………………………………………………………………..

like this,

when |A|=n, there will be ⁿC_n x 2^n-n pairs of A and B.

so total pairs will be ΣⁿCr x 2^n-r 
hence after solving this… it will be come as 3ⁿ
here n=10, so answer will be 59049.

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