Explanation
Core Mathematical Property
If a finite set contains k elements, the total number of its subsets is given by the formula:
Total Subsets=2k
Step-by-Step Derivation
Step 1: Set up the algebraic equation
The first set has m elements, so its total number of subsets is 2m.
The second set has n elements, so its total number of subsets is 2n.
According to the given problem statement, the first set has 120 more subsets than the second set:
2m−2n=120
Step 2: Factorize the equation
Since 2^m > 2^n, it implies that m > n. Let us factor out the smaller power, 2n, from the left side of the expression:
2n(2m−n−1)=120
Step 3: Prime factorize the number 120
Let us break down the number 120 into a product containing a power of 2 and an odd factor:
120=8×15
120=23×15
Step 4: Compare both sides of the equation
Now substitute this back into our equation:
2n(2m−n−1)=23×15
By comparing the even parts (powers of 2) and the odd parts on both sides:
Comparing powers of 2:
2n=23⟹n=3
Comparing the odd expressions inside the bracket:
2m−n−1=15
2m−n=15+1
2m−n=16
2m−n=24⟹m−n=4
Step 5: Solve for m
Substitute the value of n=3 into the equation m−n=4:
m−3=4
m=4+3=7
Thus, the values are m=7 and n=3.
