Question

Number of non-empty subsets of the set containing 10 elements.
(A). 1023
(B). 1024
(C). ${{2}^{10}}-2$
(D). ${{2}^{9}}$

Answer 1

Hint: To solve the above question, you need to apply the definition of the subsets and also the knowledge related selections and combinations. Focus on the point that each element in the set has 2 options, i.e., either they will be present in the subset else they will not be present.

Complete step-by-step solution -
Before starting with the solution, let us discuss different symbols and operations related to the question.
Subset: A set A is said to be the subset of set B, if all the terms of A are present in set B, i.e., set A is contained in set B. This can be represented as: $A\subset B$ .
Now let us start with the solution to the question. We know that there are 10 elements. Now let us interpret the question as: there are 2 boxes, one box contains those elements which are present in the subset, and the other contains those elements which are not present in the subset. So, each element has 2 options, one to go in the box with elements present in the subset and other to go to the other box. So, we can say that the total number of subsets are ${{2}^{10}}$ which is equal to 1024. Out of these 1024 subsets, one subset is the null set, so the number of non-empty subsets of the set containing 10 elements is 1024-1=1023.
Hence, the answer to the above question is option (c).

Note: Be careful about what is asked in the question. As the number of non-empty subsets, subsets and proper subsets are three different quantities and have different answers for a given set. As for the set in the above question, the number of non-empty subsets, subsets and proper subsets are 1023, 1024 and 1022, respectively.