Question

Let A and B be two finite sets such that there are exactly 144 sets which are subsets of A
or subsets of B. Find the number of elements in A U B.​

Answer 1

The number of elements in AUB = 11  

Given:  

A and B be two finite sets such that there are exactly 144 sets that are subsets of A or subsets of B

To find:

Find the number of elements in AUB

Solution:

Let the Number of elements in A = a

=> Number of subsets of A = 2ᵃ

And the Number of elements in B = b

=> Number of subsets of B = 2ᵇ  

Given A and B have exactly 144 subsets

=> 2ᵃ + 2ᵇ  = 144

=> 2ᵇ(2ᵃ/2ᵇ + 1) = 144

=> 2ᵇ(2ᵃ⁻ᵇ + 1) = 144  

As we know 2 power anything will be an even number then

Here 2ᵇ will be an even number and (2ᵃ⁻ᵇ + 1)  will be a odd number

So write 144 as a product of an even and odd number

=>  2ᵇ(2ᵃ⁻ᵇ + 1) = 16 × 9

=> 2ᵇ = 16

=> 2ᵇ = 2⁴     ⇒ b = 4

=> 2ᵃ⁻ᵇ + 1 =  9  

=> 2ᵃ⁻ᵇ  = 8  

=> 2ᵃ⁻ᵇ  = 2³

=> a - b = 3

=> a = 3 + 4 = 7  

From the above calculation,

=> a + b = 7 + 4 = 11  

Therefore,

The number of elements in AUB = 11  

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