Intersection of Sets - Definition & Examples

Intersection of Sets

The intersection of two (or three, or four, or more) sets contains the elements common to all of the sets. The intersection of sets is a set, itself!

The intersection of sets $A$ and $B$ (denoted as $A \cap B$) is the set containing the elements that are in both $A$ and $B!$

Here is a graphic that visually represents the intersection of sets using Venn diagrams:

Example

Let's start off with an easy example, using just two sets. Consider the following sets: $A = \{a, b, c, d\}$ and $B = \{c, d, e\}.$

To find $A \cap B,$ the intersection of sets $A$ and $B,$ we simply find the set of elements that are in both $A$ and $B.$ $$A=\{a, b, c, d\} \text{ and } B=\{c, d, e\}$$ What elements are in both $A$ and $B?$ The common elements are $c$ and $d.$ Thus, $A \cap B =\{c, d\}.$

Intersections in Set-Builder Notation

When you're learning about the intersection of two sets (let's call them $P$ and $Q$ to differentiate them from our previous example) you might encounter this set-builder notation: $$P \cap Q = \{x \: | \: x \in P \text{ and } x \in Q\}$$ Don't let this notation scare you! It simply means that the intersection of sets $P$ and $Q$ is equal to the set of all elements $x$ that are in $P$ and $Q.$

Intersection of Three Sets

You can also find the intersection of more than two sets! Consider sets $X$, $Y$, and $Z$, with $X = \{1, 2, 5, 7, 8\}, Y = \{2, 3, 5, 7, 9\},$ and $Z = \{4, 5, 7, 8, 9\}.$

To find $X \cap Y,$ the intersection of sets $X$ and $Y,$ we must find the set of elements that are in both $X$ and $Y.$ Forget about $Z$ for now. $$X = \{1, 2, 5, 7, 8\} \text{ and } Y = \{2, 3, 5, 7, 9\}$$ What elements are in both $X$ and $Y?$ The common elements are $2, 5,$ and $7.$ Thus, $X \cap Y =\{2, 5, 7\}.$

To find $X \cap Y \cap Z,$ the intersection of sets $X, Y$ and $Z,$ we must find the set of elements that are in $X, Y$ and $Z.$ $$X = \{1, 2, 5, 7, 8\}, Y = \{2, 3, 5, 7, 9\}, \text{ and } Z = \{4, 5, 7, 8, 9\}$$ What elements are in $X, Y,$ and $Z?$ The common elements, the ones in the centermost region of the Venn diagram, are $5$ and $7.$ Thus, $X \cap Y \cap Z =\{5, 7\}.$

Disjoint Sets

Sometimes sets have no elements in common, i.e., their intersection is the empty set, $\emptyset.$ When two sets have no shared elements, we say they are disjoint.

For example, consider the set of even integers and the set of odd integers. These sets are disjoint because they have no elements in common - their intersection is the empty set. Think about it: none of the even integers are also odd, and none of the odd integers are also even! These sets have no shared elements!

(Video) Sets: Union and Intersection

by patrickJMT

This video by PatrickJMT explains unions and intersections of sets.

$\large{\text{Summary}}$

The union of two sets is the set containing all the elements that appear in either of the sets. We use the symbol $\cup$. The union of set $A$ and set $B$ is the set of all the elements that are in either set $A$ $\textbf{or}$ set $B.$

An intersection of two sets is the elements that appear in both of the sets. We use the symbol $\cap$. The intersection of set $A$ and set $B$ is the set of all the elements that are in set $A$ $\textbf{and}$ set $B.$

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Suppose we had the following sets: $$A=\{a,b,c,e\}\quad B=\{b,f,g\}$$ Let's say we want to find the intersection of these sets. This intersection is represented as $A\cap B$.

Since the intersection contains all of the elements that appear in both sets, we just look at each set and list the elements that appear in both. In this case, the only element that both sets share is $b$. So, $$A\cap B=\{b\}$$

Let's look at another example. Suppose we have the following sets: $$A=\{1,3,5,7,9,\cdots\}\quad B=\{2,4,6,8,10,\cdots\}$$ What is $A\cap B$?

Since $A$ is the set of all positive odd numbers and $B$ is the set of all positive even numbers, there will be no elements that appear in both. An odd number cannot ever be even, and an even number cannot be odd. Therefore, the intersection is the empty set, the set with no elements in it. $$A\cap B=\{ \} = \emptyset$$

Intersection of Sets

Let's say we have two sets, $A$ and $B$. The intersection of sets $A$ and $B$ includes all of the values that are shared between them.

Think of it like a Venn diagram:

In the case of the Venn diagram above, set $A$ is in purple and set $B$ is in pink. The intersection of set $A$ and set $B$ is the part where these two sets overlap.

The symbol we use to show the intersection of sets is $∩$.

This looks a lot like the symbol for union. The easiest way to tell them apart is that the symbol for union ($∪$) looks like a U...like the "U" in union. The symbol for intersection ($∩$) looks like an "N", as in innnntersection.

Sometimes, you might get a problem like this:

$``x$ and $y$ are real numbers such that $\{2x + 3\} \cap \{4x -1\} = \{y\}$. Find $y."$

How do we solve this?

Since there is one value for $x,$ the evaluation of $2x + 3$ will either be exactly the same or completely different than the evaluation of $4x -1.$ In other words, after we plug $x$ into both expressions, the resulting numbers will either be the same or different.

Option A: They Evaluate to Different Numbers

Let's say that $2x+3$ evaluates to $a$ and $4x-1$ evaluates to $b$ where $a \neq b.$ Then: $$\begin{align} \{2x + 3\} \cap \{4x -1\} &= \{y\}\\ \{a\} \cap \{b\} &= \{y\}, \small{\text{ after we evaluate at $x$}}\\ \emptyset &= \{y\}, \small{(a \neq b \text{ so }\{a\} \text{ and } \{b\} \text{ have no common elements})} \end{align}$$

But remember, the empty set has no elements in it! So, if $\{y\}= \emptyset = \{ \: \},$ it must not exist! However, remember that the problem says that $y$ is a real number. Thus, $y$ cannot equal $\emptyset,$ and Option A is not possible.

Option B: They Evaluate to the Same Number

Let's say that $2x + 3$ and $4x-1$ both evaluate to $a,$ which will be a real number. Then: $$\begin{align} \{2x + 3\} \cap \{4x -1\} &= \{y\}\\ \{a\} \cap \{a\} &= \{y\}, \small{\text{ after we evaluate at $x$}}\\ \{a\} &= \{y\}\\ a&= y \end{align}$$ Here, $y=a,$ a real number! Option B works!

Solving the Problem

Now we know that we need $2x + 3$ and $4x-1$ to evaluate to the same number. In other words, to solve the problem, we need to set $2x + 3$ and $4x-1$ equal to each other. Then, we'll solve for $x$, and finally we'll use our knowledge of intersection of sets to solve for $y.$ Here's how it works:

Step 1: Solve for $x$

$$\begin{align} 2x+3 &= 4x-1\\ 2x\,{\color{#8925AE}{-\,4x}}+3&=4x\,{\color{#8925AE}{-\,4x}}-1\\ -2x+3&=-1\\ -2x+3\,\color{#8925AE}{-\,3}&=-1\,\color{#8925AE}{-\,3}\\ -2x&=-4\\ \frac{-2x}{\color{#8925AE}{-2}}&=\frac{-4}{\color{#8925AE}{-2}}\\ x&=2 \end{align}$$

Step 2: Plug in $x=2$ to solve for $y:$

$$\begin{align} \{2x + 3\} \cap \{4x -1\} &= \{y\}\\ \{2({\color{#8925AE}{2}}) + 3\} \cap \{4({\color{#8925AE}{2}}) -1\} &= \{y\}\\ \{4 + 3\} \cap \{8 -1\} &= \{y\}\\ \{7\} \cap \{7\} &= \{y\}\\ \{7\} &= \{y\}\\ 7&=y \end{align}$$

So, $y=7.$ Let's try another one to practice!

We can solve this problem exactly the way we did before.

Step 1: Solve for $x$ $$\begin{align} 5x+15&= 7x+7\\ 5x\,{\color{#8925AE}{-\,7x}}+15&=7x\,{\color{#8925AE}{-\,7x}}+7\\ -2x+15&=7\\ -2x+15\,\color{#8925AE}{-\,15}&=7\,\color{#8925AE}{-\,15}\\ -2x&=-8\\ \frac{-2x}{\color{#8925AE}{-2}}&=\frac{-8}{\color{#8925AE}{-2}}\\ x&=4 \end{align}$$

Step 2: Plug in $x=4$ to solve for $y:$ $$\begin{align} \{5x+15\} \cap \{7x+7\} &= \{y\}\\ \{5({\color{#8925AE}{4}})+15\} \cap \{7({\color{#8925AE}{4}})+7\} &= \{y\}\\ \{20+15\} \cap \{28+7\} &= \{y\}\\ \{35\} \cap \{35\} &= \{y\}\\ \{35\} &= \{y\}\\ 35&=y \end{align}$$ So, $y=35$.

The intersection of sets is the elements that they share

$A \cap B$ is read as "$A$ intersection $B$". The intersection of sets $A$ and $B$ contains all the elements that are in both $A$ and $B$. In other words, all the elements in the intersection $A \cap B$ will also be elements of both of $A$ and $B$.

In set builder notation, we can say that: $A \cap B = \{x \mid x \in A$ and $x \in B\}$.

Examples:

$$\{1, 2, 3\} \cap \{3, 4\} = \{3\}$$

$3$ is in both the set before the $\cap$ and the set after it, so it's in the intersection. Since no other elements overlap, $3$ is the only thing in the intersection. $$\{3, 7, 9\} \cap \{\{2, 3\}, 4\} = \emptyset$$ The two sets have no elements in common, so their intersection is the empty set, $\emptyset,$ or the set with no elements in it. The empty set can also be represented like this: $\{\: \}.$ $$\{1, 2, 3\} \cap \{1, 2, 3\} = \{1, 2, 3\}$$ This is a universal set because it contains all of the elements.

$$\{5, 6, 2, \{1, 2\}\} \cap \{5, 6, 2, \{1, 2\}\} = \{5, 6, 2, \{1, 2\}\}$$

Both sets there are identical, so their intersection will also be identical to them (since that means that any element that's in one is in both of them).

$$\{1, 3, 5\} \cap \{9\} = \emptyset$$

$9$ isn't anywhere in the first set, so we know there's no overlap between the two. That means that the intersection has no elements in it, which means it's the empty set ($\emptyset$).