Question

The number of non-empty subsets of the set {1, 2, 3, 4} on
A. 14
B. 15
C. 16
D. 17

Answer 1

Hint: We know that a set is a collection of well defined objects i.e the objects in a set follow a given rules. Therefore the set given above is a collection of first four natural numbers. The members of a set are called its elements and 1, 2, 3 and 4 are the elements of the given set.

Complete step-by-step answer:
Sets can be of different types. The above set is a finite set because it contains only a finite number of elements and we know that there are exactly four elements in the set.
We have to find the subsets of the given set which must be non empty in nature which means that none of the elements must not be null or φ. By a subset we mean that each element of one set is also an element of another set then the first set is a subset of another set and it can be written as $ A \subset B $ where if we suppose one set to be A the other will be set B, and for A to be the subset of B elements of A must be in B.
Therefore the subset set will have the following elements:
 $ \Rightarrow $ {{1}, {2}, {3}, {4}, {1,1}, {1,2}, {1,3}, {1,4}, {2,1}, {2,2}, {2,3}, {2,4}, {3,1}, {3,2}, {3,3}, {3,4}, {1,2,3}, {1,2,4}, {1,3,4}, {2,3,4}, {1,2,3,4}}
Now all these non empty subsets can be counted. There is an alternative way to it as well
The number of elements in the subset is given by $ {2^n} $ where n is the number of elements of the set. Since we have to find only non empty elements hence it can be given by,
 $ \Rightarrow {2^n} - 1 = {2^4} - 1 = 16 - 1 = 15 $
So, the correct answer is “15”.

Note: When A is a subset of B then all the elements of A must belong to B but one must not get confused and conclude that all the elements of B belong to A as well , for that to be possible both the sets A and B must be subset of each other.