Subsets, Equality, and Power Sets

From Elements to Collections

In the previous lesson, we asked questions such as

That compares an object with a set.

Now suppose

and

Every element of \(B\) is also an element of \(A\). We therefore say that \(B\) is a subset of \(A\).

This introduces a new kind of relationship:

That distinction matters. Membership and subset relationships are closely related, but they do different jobs.

What Is a Subset?

Let \(A\) and \(B\) be sets.

We say that \(A\) is a subset of \(B\) when every element of \(A\) is also an element of \(B\).

This can be stated precisely as:

For example, let

and

The elements of \(A\) are \(1\) and \(3\). Both occur in \(B\). Therefore,

A Simple Subset Test

To determine whether

ask:

Can I find any element of \(A\) that is missing from \(B\)?

If the answer is no, then \(A\subseteq B\).

If even one element of \(A\) is absent from \(B\), then \(A\) is not a subset of \(B\).

For example,

because \(5\) belongs to the first set but not the second.

Membership versus Subset

This is one of the most important notation distinctions in elementary set theory.

Let

Then

because the number \(2\) is an element of \(A\).

But

because every element of the set \(\{2\}\) is also an element of \(A\).

Compare the two:

means

“The object \(2\) belongs to \(A\).”

while

means

“The set \(\{2\}\) is contained within \(A\).”

A useful mental check is:

Nested Sets Can Change the Answer

Suppose instead that

Now

The set \(\{2\}\) itself appears as an element of \(B\).

But

The braces matter because they tell us what the actual mathematical object is.

Set Equality

Two sets are equal when they contain exactly the same elements.

If

and

then

Order does not matter in a set.

Likewise,

Repeating an element does not change the set.

Equality through Subsets

There is a powerful way to characterize set equality:

exactly when

and

In words, if every element of \(A\) belongs to \(B\), and every element of \(B\) belongs to \(A\), then the two sets must contain exactly the same elements.

The symbol \(\iff\) means “if and only if.”

We will encounter this style of reasoning repeatedly in mathematics: instead of comparing complicated objects all at once, we prove that each one is contained in the other.

Proper Subsets

Sometimes one set is contained inside another but is not equal to it.

For example,

and

We have

but

In this case, \(A\) is called a proper subset of \(B\). We will write

The notation \(\subsetneq\) tells us both that

and that

Why Use \(\subsetneq\)?

Some textbooks use \(\subset\) for proper subset, while others use it more loosely for subset. To avoid ambiguity, Malone Math will use

for ordinary subset and

when we specifically mean a proper subset.

The Empty Set as a Subset

The empty set,

contains no elements.

An important result is:

for every set \(A\).

At first this can feel strange. How can a set containing nothing be inside every set?

Remember the definition of subset:

Every element of the first set must also belong to the second set.

But the empty set has no elements that can violate this condition. There is nothing in \(\varnothing\) that fails to belong to \(A\).

Therefore,

for every possible set \(A\).

A Logical View

This is an early example of what is sometimes called vacuous truth.

The statement

“Every element of the empty set belongs to \(A\)”

is true because there are no counterexamples.

We do not need to explore formal logic deeply yet, but this connection will be useful later.

Every Set Is a Subset of Itself

For any set \(A\),

Why?

Because every element of \(A\) is certainly an element of \(A\).

Therefore, for any set \(A\), we automatically know two subsets:

and

These are sometimes called the improper subsets of \(A\), while the remaining subsets are proper subsets.

Power Sets

Now we can take a major conceptual step.

Instead of asking whether one particular set is a subset of another, we can collect all possible subsets of a set into a new set.

This new collection is called the power set.

The power set of \(A\) is written

Let

The subsets of \(A\) are

Therefore,

Notice the Levels of Braces

The elements of \(\mathcal{P}(A)\) are themselves sets.

For example,

That statement is true because \(\{a\}\) is one of the subsets of \(A\).

At the same time,

because \(a\) itself is not a subset of \(A\). The set \(\{a\}\) is.

Power sets are an excellent place to practice distinguishing membership from subset relationships.

Counting Subsets

Suppose a set contains \(n\) elements.

For every element, we have exactly two choices when building a subset:

• include the element, or
• do not include the element.

If the set contains one element, there are

possible subsets.

If it contains two elements, there are

possible subsets.

If it contains three elements, there are

In general, if

then

Here \(|A|\) means the cardinality of \(A\): the number of elements in the set.

Example

Let

Because

the number of subsets is

Indeed,

The Binary Connection

There is a useful connection hiding inside the formula \(2^n\).

For each element, we make a yes-or-no decision:

For the set

a choice such as

could mean:

• include \(a\),
• exclude \(b\),
• include \(c\).

That binary pattern corresponds to the subset

So power sets connect naturally with binary strings, combinatorics, digital logic, and computer science.

Why Subsets Matter

Subsets are not merely an exercise in arranging braces. They model a very common question:

Which selections can be made from a larger collection?

Permissions

Suppose a software system has three optional permissions:

Any particular user's permission package is a subset of \(P\).

The power set \(\mathcal{P}(P)\) describes every possible combination of permissions.

Feature Selection

If a product has optional features

each possible configuration corresponds to a subset of \(F\).

There are

possible feature combinations.

Probability

If a sample space is

then an event such as

is a subset of the sample space.

Probability theory therefore uses subsets constantly: events are collections of possible outcomes.

Solutions and Domains

A solution set is normally a subset of some larger number system. For example,

is a subset of the real numbers:

Likewise, the domain of a function may be a particular subset of \(\mathbb{R}\), \(\mathbb{Z}\), or some other set.

Worked Examples

Example 1: Testing a Subset

Let

and

Every element of \(A\) appears in \(B\), so

Because \(A\neq B\), we can say more specifically

Example 2: Finding a Failed Subset

Let

and

The number \(6\) belongs to \(C\), but

Therefore,

Only one missing element is needed to show that a subset relationship fails.

Example 3: Proving Equality

Suppose

and

Every member of \(A\) belongs to \(B\), so

Every member of \(B\) also satisfies the rule defining \(A\), so

Therefore,

Example 4: Constructing a Power Set

Let

Start with the empty selection:

Add the one-element selections:

Add the selection containing both elements:

Therefore,

Example 5: Counting without Listing

A set contains \(7\) elements. How many subsets does it have?

Use

Therefore,

The set has

different subsets.

Common Errors

1. Confusing \(\in\) with \(\subseteq\)

If

then

but

The first compares an element with a set. The second compares two sets.

2. Forgetting the Empty Set

When listing all subsets of a set, always include

The empty set is a subset of every set.

3. Forgetting the Whole Set

Every set is a subset of itself:

Therefore \(A\) itself must appear in \(\mathcal{P}(A)\).

4. Confusing a Power Set with the Original Set

If

then

The power set contains sets as its elements.

5. Assuming a Proper Subset Can Equal the Original Set

If

then necessarily

If the sets are equal, the relationship is a subset relationship but not a proper-subset relationship.

Practice

Let

Determine whether each statement is true or false.

1. \(\{1,3\}\subseteq A\)
2. \(\{1,5\}\subseteq A\)
3. \(2\in A\)
4. \(\{2\}\in A\)
5. \(\varnothing\subseteq A\)
6. \(A\subseteq A\)
7. \(\{1,2\}\subsetneq A\)

Power Set Practice

Find the power set of

Counting Practice

A system has \(10\) independently selectable options. If every combination of selected options is allowed, how many different selections are possible?

The Deeper Idea

Membership lets us ask whether an object belongs to a collection.

Subset language moves one level higher and asks whether an entire collection fits inside another collection.

The power set moves one level higher again by collecting all those possible subsets into a new set:

So there is a natural progression:

This ability to organize mathematical objects into layers of structure will become important when we study relations, functions, probability, combinatorics, and more advanced mathematics.

Summary

• \(A\subseteq B\) means every element of \(A\) is also an element of \(B\).
• \(A\nsubseteq B\) means at least one element of \(A\) does not belong to \(B\).
• Membership, \[ x\in A, \] compares an element with a set.
• Subset notation, \[ A\subseteq B, \] compares one set with another.
• Two sets are equal exactly when they contain the same elements: \[ A=B \iff A\subseteq B \text{ and } B\subseteq A. \]
• A proper subset satisfies \[ A\subsetneq B, \] meaning \(A\subseteq B\) but \(A\neq B\).
• The empty set is a subset of every set: \[ \varnothing\subseteq A. \]
• Every set is a subset of itself: \[ A\subseteq A. \]
• The power set \[ \mathcal{P}(A) \] is the set containing every subset of \(A\).
• If \[ |A|=n, \] then \[ |\mathcal{P}(A)|=2^n. \]

Subsets give us a systematic way to talk about selecting, containing, and organizing mathematical objects. That language will become especially important as we begin building relations and functions from sets.

Sets, Relations, and Functions

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