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 3 elements in our original set. We can use each of these to form the single element subsets.
{4}, {8}, {12}
Two-Elements Subsets
Next, we need to find all the two-elements subsets.
{ 4, 8 }, { 8, 12 }, { 4, 12 }
Three-Elements Subsets
The original set has 3 elements, so we can only form one subset having 3 elements, the original set itself.
{ 4, 8, 12 }
Summary
We have a total of 8 subsets.
& ∅, {4}, {8}, {12},
&{ 4, 8 }, { 8, 12 }, { 4, 12 }, { 4, 8, 12 }
Extra
Using Set-Builder Notation
Let's recall what a set-builder notation is and try to write the given set in this notation. Set-builder notation describes characteristic of elements in the set. Let's look at an example.
{x | xis a natural number, x<4} = {1,2,3}
We read this notation as the set of natural numbers x, such as x is less than 4. Now, let's focus on our set. {4,8,12}
Notice that all numbers are multiples of 4 and are less than or equal to 12. This means that we can describe all elements in the set.
{x | xis a multiple of4, x≤ 12} = {4,8,12}
