Question

How many proper subsets does the set Y have if │Y│= k? - Discrete Math

Answer 1

What is a proper sub-set?

A proper subset of a set S, is a subset of S that is not equal to S. In other words, if R is a subset of S, then all the elements of R are in S but S contains atleast one element that is not in R.

Example:

If S = {1,2,3}, R = {1,3} -> R is a proper subset of S

In our given question, set Y contains k distinct elements.

The total number of subsets of any set is given by where n is the number of elements in the set. In our question , as the set contains k elements. This is usually referred to as power set. Power set contains all the subsets including the set itself. We know that every set is a subset of itself.

But for our solution we are only interested in the number of proper subsets. For this we have to remove one from the total number of sets.

So our answer for the given question would be

Example:

Y = {1,2,3}

Here k = 3

All subsets of Y = { {1}, {2}, {3}, {1,2} ,{1,3}, {2,3}, {1,2,3}, {null} } . The total number of subsets is 8 which is equal to

Proper subsets of Y = { {1}, {2}, {3}, {1,2} ,{1,3}, {2,3}, {null} }. The total number of subsets is 8 which is equal to