Concept:A proper subset of a set is any subset that is not equal to the original set itself.
A superset of a given set must contain all elements of that given set.
Explanation:We need to count all proper subsets of
{1,2,3,4} that also contain the element
3.
First, find all supersets of
{3} inside
{1,2,3,4}.
Any superset of
{3} is formed by taking
{3} together with any subset of the remaining elements
{1,2,4}.
The set
{1,2,4} has
23=8 subsets.
Each such subset, when unioned with
{3}, gives a distinct superset of
{3}.
These
8 supersets include the full set
{1,2,3,4} itself.
Since we only want
proper subsets, we exclude
{1,2,3,4}.
Therefore, the count is
8−1=7.
The seven proper subsets are:
{3},{1,3},{2,3},{3,4},{1,2,3},{1,3,4},{2,3,4}.
Answer:Option B. 7