Two finite sets have ' m ' and ' n ' number of elements respectively. The total number of subsets of the first set is 112 more than the total number of subsets of the second set. Then the values of m and n are respectively.

Step-by-step solution

The number of subsets of a set with m elements is 2^ m . The number of subsets of a set with n elements is 2^ n . Given the condition 2^ m - 2^ n = 112 . Factoring out 2^ n , we get 2^ n (2^ m-n - 1) = 112 . Expressing 112 as a product of a power of 2 and an odd number: 112 = 16 7 = 2⁴ 7 . Comparing 2^ n (2^ m-n - 1) = 2⁴ 7 , we have 2^ n = 2⁴ , which implies n = 4 . Also, 2^ m-n - 1 = 7 , which implies 2^ m-n = 8 = 2³ . Thus, m - n = 3 . Substituting n = 4 , we get m - 4 = 3 , so m = 7 . The values are m = 7 and n