A subset of a set is also a set. The requirement is that all elements in the subset are also elements of the set. We can list all the subsets starting with the ones having 0 elements, followed by 1, 2, and 3 elements.
Zero-Element Subsets
The only subset of any set with zero elements is the empty set. We write this using the empty set notation.
∅
One-Element Subsets
There are three elements in our original set. We can use each of these to form the single element subsets.
{dog}, {cat}, {fish}
Two-Element Subsets
Next, we need to find all the two-element subsets.
{dog, cat}, {dog, fish}, {cat, fish}
Three-Element Subsets
The original set has three elements, so we can only form one subset having three elements, the original set itself.
{dog, cat, fish}
Summary
We have a total of 8 subsets.
∅, {dog}, {cat}, {fish},
{dog, cat}, {dog, fish}, {cat, fish},
{dog, cat, fish}
