Question

two finite sets have m and n elements. The total number of subsets of first set is 240 more than the subset of second set . Find the value of m and n​

Answer 1

Step-by-step explanation:

Given :-

Two finite sets have m and n elements. The total number of subsets of first set is 240 more than the subset of second set .

To find :-

Find the value of m and n ?

Solution :-

Let the two sets be A and B

Let the first set be A

Let the second set be B

Let the set A having m elements

n(A) = m

Let the set B having n elements

n(B) = n

We know that

If a set having p elements in the set then the number of subsets to the set = 2^p

The number of subsets in the set A = 2^m

The number of subsets in the set B = 2^n

Given that

The total number of subsets of first set is 240 more than the subset of second set

=> 2^m = 2^n +240

=> 2^m - 2^n = 240

On multiplying the numerator and the denominator with 2^n in LHS

=> 2^n [2^m - 2^n ] / 2^n = 240

=> 2^n [ (2^m)/2^n - (2^n/2^n) ] = 240

=>2^n [ (2^m / 2^n ) - 1 ] = 240

=> 2^n [ 2^(m-n) - 1 ] = 240

Since a^m × a^n = a^(m+n)

=> 2^n [ 2^(m-n) - 1 ] = 2×2×2×2×3×5

=> 2^n [ 2^(m-n) - 1 ] = 2⁴×3¹×5¹

=> 2^n [ 2^(m-n) - 1 ] = 2⁴ × 15

On Comparing both sides then

=> 2^n = 2⁴

If the bases are equal then exponents must be equal.

=> n = 4

and

2^(m-n) - 1 = 15

=> 2^(m-4) - 1 = 15

=> 2^(m-4) = 15+1

=> 2^(m-4) = 16

=> 2^(m-4) = 2×2×2×2

=> 2^(m-4) = 2⁴

If the bases are equal then exponents must be equal.

=> m-4 = 4

=> m = 4+4

=> m = 8

Therefore, m = 8 and n = 4

Answer:-

The values of m and n for the given problem are 8 and 4 respectively.

Check :-

n(A) = m = 8

Number of sub sets of A = 2⁸ = 256

n(B) = n = 4

Number of subsets of B = 2⁴ = 16

Their difference = 256-16 = 240

Verified the given relations in the given problem.

Used formulae:-

• If a set having m elements in the set then the number of subsets to the set = 2^m

• a^m × a^n = a^(m+n)

• If the bases are equal then exponents must be equal.

• If a^m = a^n => m = n

Answer 2

Let the first set = A

number of subset = 2m

Let the second set be=B

number of subset=2n

$∵2m−2n=112$

$2n(2m−n−1)=128−16$

$2n(2m−n−1)=24(23−1)$

$Equaten=4$

$m−n=3m=n \\ +3m=4+3=7 \\ \\ ∴m×n=4×7=28$

Hope it helps~

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