Question

Two finite sets have m and n elements. The total number of subsets of the first set is 112 more than the total number of subsets of the second set. Find the valuesof m and n​

Answer 1

Number of subsets of the two sets = 2m and 2n, respectively (Formula)

According to the question,

2m = 2n – 112

ð 2m - 2n = 112

ð 2n (2m / 2n - 1) = 24 (7)

ð 2n ( 2m - n - 1) = 24 ( 7 )

2n cannot be equal to 7 (since 2n = 2 x 2 x 2 x 2 x .........)

Therefore, 2n = 24

Equating the powers of 2, we get:

n = 4.......... (1)

Also, 2m - n -1 = 7

ð 2m - n = 7+1 = 8

ð 2m - n = 23 (since 8 = 23)

Equating the powers of 2, we get:

m - n= 3

ð m = n+3 = 4+3 ( From (1) )

ð m = 7

Answer 2

Answer;

Number of subsets of the two sets = 2m and 2n, respectively (Formula)

According to the question,

2m = 2n – 112

ð 2m - 2n = 112

ð 2n (2m / 2n - 1) = 24 (7)

ð 2n ( 2m - n - 1) = 24 ( 7 )

2n cannot be equal to 7 (since 2n = 2 x 2 x 2 x 2 x .........)

Therefore, 2n = 24

Equating the powers of 2, we get:

n = 4.......... (1)

Also, 2m - n -1 = 7

ð 2m - n = 7+1 = 8

ð 2m - n = 23 (since 8 = 23)

Equating the powers of 2, we get:

m - n= 3

ð m = n+3 = 4+3 ( From (1) )

ð m = 7