Question

Two finite sets have m and a elements respectively. The total number of subsets of the first set is 56 more than the total number of subsets of the second set. The values of m and a respectively are:

• A. 7, 6
• B. 5, 1
• C. 6, 3
• D. 8, 7

Answer 1

The Correct Option is C

Let the first set have \( m \) elements, and the second set have \( a \) elements. The number of subsets of a set with \( n \) elements is given by \( 2^n \).

The number of subsets of the first set is \( 2^m \).
The number of subsets of the second set is \( 2^a \).

We are given that the total number of subsets of the first set is 56 more than the total number of subsets of the second set. This can be written as:

\[ 2^m = 2^a + 56 \]

We can rewrite this equation as:

\[ 2^m - 2^a = 56 \]

We can factor out \( 2^a \):

\[ 2^a (2^{m-a} - 1) = 56 \]

Since \( 56 = 2^3 \times 7 \), we can consider the factors of 56 which are powers of 2. The factors are 1, 2, 4, 8, 14, 28, 56.
We can write 56 as \( 2^a \times k \), where \( k \) is an integer.
Then \( 2^{m-a} - 1 = k \).

If \( 2^a = 8 \), then \( a = 3 \).
In this case, \( 2^{m-3} - 1 = 7 \), which gives \( 2^{m-3} = 8 = 2^3 \).
Thus, \( m - 3 = 3 \), so \( m = 6 \).

If \( 2^a = 1 \), then \( a = 0 \). This gives \( 2^m - 1 = 56 \), so \( 2^m = 57 \), which is not possible since \( 2^m \) must be a power of 2.
If \( 2^a = 2 \), then \( a = 1 \). This gives \( 2^{m-1} - 1 = 28 \), so \( 2^{m-1} = 29 \), which is not possible.
If \( 2^a = 4 \), then \( a = 2 \). This gives \( 2^{m-2} - 1 = 14 \), so \( 2^{m-2} = 15 \), which is not possible.
If \( 2^a = 28 \), then this is not a power of 2.
If \( 2^a = 56 \), then this is not a power of 2.

Therefore, the only solution is \( a = 3 \) and \( m = 6 \).

Final Answer: The final answer is \( {m=6, a=3} \).

Answer 2

Given:

Two finite sets have m and n elements respectively.

The total number of subsets of the first set is 56 more than the total number of subsets of the second set.

Step 1: Understand the number of subsets of a set.

A set with k elements has \(2^k\) subsets (including the empty set and the set itself).

Step 2: Translate the given condition into an equation.

The first set has mm elements, so its total number of subsets is \(2^m\)

The second set has n elements, so its total number of subsets is 2n

According to the problem, the first set has 56 more subsets than the second set:

• \(2^m=2^n+56\)

Step 3: Test the given options to find mm and nn that satisfy the equation.

Option (C): m=6, n=3

\(2^6=2^3+56\)
64=8+56
64 = 64(True)

This satisfies the equation perfectly.

Verification of other options:

Option (A): m=7, n=6

• \(2^7=2^6+56\)
  128=64+56
  128=120(False)

Option (B): m=5, n=1

• \(2^5=2^1+56\)
  32=2+56
  32=58(False)

Option (D): m=8, n=7

• \(2^8=2^7+56\)
  256=128+56
  256=184(False)