Question

EXAMPLE 2. Two finite sets having m and n elements,
The total number of subsets of the first set is 56 more
than the total number of subsets of the second set. Find
the values of m and n.
a. 6,5 b. 6,3
c. 6,6
d. 5,4
Plz.. Explain it​

Answer 1

Answer:

(b) 6, 3

Step-by-step explanation:

Suppose a set consists of n elements.

Either an element can exist or cannot exist in a subset. So, if an element is present, we can represent it as 1, and if it is absent, we can represent it as 0. Thus, an n-bit binary string is formed.

Each place can consist of either 0 or 1. So, there are 2 possibilities. Since there are n such places, therefore, there are

2×2×2×... n times possible binary strings

=$2^n$ possible subsets or binary strings

Now, coming back to the question.

It is given that total number of subsets of the first set is 56 more than the total number of subsets of the second set.

∴ $2^m -2^n = 56$

Since m>n, so, we can write $m$ as $n+x$.

So, the equation becomes

$2^\[n+x$ - $2^n$ $= 56$

or, $2^n(2^x-1)=56$

Now, we need to think about factors of 56.

56=1×56=2×28=4×14=7×8

Among these, we need to pick a pair of factors such that one of them is a power of 2 and another is one less than a power of two (which is also an odd number).

Since, a number one less than a power of 2 (which is an odd number) occurs in the factorization 56=7×8, so

($2^x$ - 1) = 7 and  $2^n$= 8.

Solving the above two equations, we get $m=3$ and $x=3$.

∴$m = n+x = 6$

Hence, the values of m and n are 6 and 3 respectively.