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.

\emptyset

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.

\emptyset, {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 ∣ x is 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 ∣ x is a multiple of 4, x \leq 12} = {4, 8, 12}

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