Union, Intersection, Difference, and Complement

Doing Operations with Sets

Numbers have operations such as addition and multiplication. Sets have operations too.

Suppose

and

We might ask:

• Which elements occur in either set?
• Which elements occur in both sets?
• Which elements occur in \(A\) but not \(B\)?
• Which allowed elements are outside \(A\)?

Those questions lead respectively to union, intersection, difference, and complement.

The notation is new, but the ideas are familiar: combine, overlap, remove, and exclude.

Union

The union of two sets contains every element belonging to the first set, the second set, or both.

The symbol for union is

so we write

If

and

then

The repeated elements \(3\) and \(4\) are written only once because sets contain distinct elements.

Membership Definition

More precisely,

when

Here or is inclusive: an element may belong to \(A\), to \(B\), or to both.

A useful verbal shortcut is:

Intersection

The intersection of two sets contains only the elements that belong to both sets.

The symbol is

so we write

Using

and

we obtain

These are the elements shared by the two sets.

Membership Definition

exactly when

A useful verbal shortcut is:

Disjoint Sets

If two sets have no common elements, their intersection is empty.

For example,

Sets with no common elements are called disjoint sets.

Set Difference

The difference of two sets removes from the first set everything that belongs to the second.

We may write

If

and

then

We begin with \(A\) and remove the elements that also occur in \(B\).

In the opposite direction,

Therefore, in general,

Membership Definition

exactly when

A useful verbal shortcut is:

This Is Not Numerical Subtraction

The word difference is also used for subtraction, but set difference does not subtract the numerical values of elements.

For example,

not some arithmetic result involving \(2-5\) or \(8-5\).

Complement

A complement describes everything under consideration that does not belong to a particular set.

For this operation, we must know the larger collection in which we are working. This larger set is often called the universal set and written \(U\).

Suppose

and

The complement of \(A\) is

Equivalently,

Membership Definition

Assuming \(x\in U\),

exactly when

A useful verbal shortcut is:

The Universal Set Matters

A complement cannot be interpreted correctly without knowing what universe of objects is being considered.

For example, if

then its complement will be different depending on whether the universal set is

or

Complement is therefore always relative to a larger set.

Set Operations and Logic

Set operations connect directly with logical statements.

Suppose membership in \(A\) means that condition \(P\) is true, while membership in \(B\) means that condition \(Q\) is true.

Then:

corresponds to

while

corresponds to

Likewise,

corresponds to

And

corresponds to

This is an important bridge between set theory and logic:

The same structure will appear later in inequalities, probability, computer logic, database searches, and programming conditions.

Set Operations with Intervals

Intervals on the number line are sets, so the same operations apply to them.

Let

and

Union

Together, the intervals cover every number from \(-2\) through \(7\):

Intersection

The overlap runs from \(1\) through \(4\):

Difference

The portion of \(A\) not contained in \(B\) is

Notice that \(1\) is excluded because \(1\in B\).

Complement

If the universal set is \(\mathbb{R}\), then

This connects set notation directly with the interval notation used for inequalities and domains.

Laws of Set Operations

Set operations obey structural rules similar to the properties of arithmetic.

Commutative Laws

The order of two sets does not matter for union or intersection.

Associative Laws

Grouping does not matter when we repeatedly use the same operation.

Distributive Laws

and

Notice something interesting: union distributes over intersection, and intersection distributes over union.

Idempotent Laws

Combining a set with itself does not create anything new.

Identity Laws

Complement Laws

These laws are useful because set expressions can be manipulated and simplified much like algebraic expressions.

De Morgan's Laws

Complementing a union or intersection produces an important pair of rules known as De Morgan's laws.

and

The first says:

To be outside \(A\cup B\), an element must be outside \(A\) and outside \(B\).

The second says:

To be outside \(A\cap B\), an element only needs to be outside \(A\) or outside \(B\).

A Familiar Language Example

Suppose a student must be both

• at least \(18\) years old, and
• enrolled in the course.

Failing that combined requirement means the student is

• not at least \(18\), or
• not enrolled,

or both.

That is the logical structure behind

De Morgan's laws therefore belong not only to set theory but also to logic, programming, probability, and circuit design.

Counting a Union

Suppose \(A\) and \(B\) are finite sets.

A first instinct might be to count the elements of both sets by adding

But if the sets overlap, elements in the intersection have now been counted twice.

To correct that duplication, subtract the intersection once:

This is the simplest form of the inclusion-exclusion principle.

Example

Suppose \(25\) students study French, \(18\) study Spanish, and \(7\) study both.

The number studying at least one of the languages is

Therefore,

We added both groups and then removed one copy of the students who had been counted twice.

Practical Applications

Database Searches

Suppose a company database contains the sets

and

Then

represents customers who are both in Washington and actively subscribed.

Meanwhile,

represents Washington customers without an active subscription.

Scheduling

If \(A\) is the set of times you are available and \(B\) is the set of times another person is available, then

contains the possible meeting times.

Probability

If \(A\) and \(B\) are events, then

means that at least one of the events occurs, while

means that both occur.

The complement

means that event \(A\) does not occur.

Solving Inequalities

Consider

The solution set is the union

Set operations give us a precise language for combining separate parts of a solution.

Computer Permissions

Suppose \(A\) contains users permitted to read a file and \(B\) contains users permitted to edit it.

Then

contains users with both permissions, while

contains users who may read but not edit.

Worked Examples

Example 1: Four Operations

Let

and

The union is

The intersection is

The difference is

In the other direction,

Example 2: Finding a Complement

Let

and let

Then

Example 3: Three Sets

Let

Evaluate

First find

Then intersect with \(A\):

We could also use the distributive law:

Since

and

we get

Example 4: Set Difference with Intervals

Let

and

Then

The endpoints \(4\) and \(7\) are removed because they belong to \(B\).

Example 5: Inclusion-Exclusion

A workplace survey finds that \(60\) employees use spreadsheet software, \(45\) use database software, and \(20\) use both.

The number who use at least one is

Therefore,

Common Errors

1. Counting Shared Elements Twice in a Union

If

and

then

not

2. Confusing Union and Intersection

Remember:

while

3. Reversing Set Difference

and

are generally different.

Always begin with the set written first.

4. Taking a Complement without a Universe

The statement \(A^c\) only has meaning relative to some universal set \(U\).

5. Treating “Or” as Exclusive

In the definition of union, “\(A\) or \(B\)” includes elements that belong to both.

6. Forgetting Endpoint Membership

When using intervals, decide carefully whether each endpoint remains after a union, intersection, or difference.

Practice

Let

and

Find:

1. \(A\cup B\)
2. \(A\cap B\)
3. \(A\setminus B\)
4. \(B\setminus A\)
5. \(A^c\)
6. \((A\cup B)^c\)

Interval Practice

Let

and

Find:

1. \(A\cup B\)
2. \(A\cap B\)
3. \(A\setminus B\)

Counting Practice

At a school, \(42\) students participate in music, \(31\) participate in sports, and \(14\) participate in both.

How many participate in at least one of the two activities?

The Deeper Idea

Set operations give us a compact way to build complicated conditions from simple ones.

If \(A\) represents one condition and \(B\) another, then

means one condition or the other,

means both conditions,

means the first condition without the second, and

means the first condition is not satisfied.

That gives us a reusable mathematical language:

As mathematics becomes more advanced, this same toolkit will continue appearing in logic, probability, functions, topology, databases, combinatorics, and mathematical analysis.

Summary

• The union \[ A\cup B \] contains elements belonging to \(A\), \(B\), or both.
• The intersection \[ A\cap B \] contains elements belonging to both sets.
• The difference \[ A\setminus B \] contains elements of \(A\) that are not in \(B\).
• The complement \[ A^c \] contains elements of the universal set \(U\) that are not in \(A\).
• If \[ A\cap B=\varnothing, \] then \(A\) and \(B\) are disjoint.
• Union corresponds naturally to logical or.
• Intersection corresponds naturally to logical and.
• Complement corresponds naturally to logical not.
• De Morgan's laws are \[ (A\cup B)^c=A^c\cap B^c \] and \[ (A\cap B)^c=A^c\cup B^c. \]
• For finite sets, \[ |A\cup B| = |A|+|B|-|A\cap B|. \]

These operations turn sets from passive collections into working mathematical tools. They let us combine conditions, isolate overlaps, remove unwanted cases, and describe everything outside a chosen set.

Sets, Relations, and Functions

• Previous: Subsets, Equality, and Power Sets
• Sets, Relations, and Functions — return to the lesson directory.