Venn diagrams

Venn diagrams are simple pictures that help us see how sets are related to each other. In this chapter we focus on how to read, draw, and use Venn diagrams for a small number of sets, usually two or three.

We will assume you already know what sets and elements are, and that you have seen basic set notation (such as $A$, $B$, $A \cup B$, $A \cap B$) in other chapters. Here we use Venn diagrams mainly as a visual aid for these ideas.

Basic structure of a Venn diagram

A typical Venn diagram has:

• A rectangle representing the universal set $U$ (everything under consideration in the situation).
• One or more overlapping shapes (usually circles), each representing a set.
• Regions formed by overlaps representing various combinations (like “in $A$ but not in $B$”, “in both $A$ and $B$”, etc.).

For example, with two sets $A$ and $B$:

• The rectangle is $U$.
• One circle is $A$.
• The other circle is $B$.
• Where the circles overlap is $A \cap B$ (elements in both $A$ and $B$).
• Parts of circles not overlapping represent elements in exactly one of the sets.

The picture does not need to be drawn perfectly; what matters is keeping track of which region represents which part of the sets.

Two-set Venn diagrams

With two sets $A$ and $B$ inside the universal set $U$, the diagram is divided into four possible regions:

1. In $A$ only (in $A$ but not in $B$).
2. In $B$ only (in $B$ but not in $A$).
3. In both $A$ and $B$ (the overlap).
4. In neither $A$ nor $B$ (in $U$ but outside both circles).

It is important to think in terms of these regions when reading or filling in a diagram.

Showing union and intersection

Venn diagrams are especially useful to see unions and intersections:

• $A \cup B$ (union) is everything in at least one of $A$ or $B$. On the diagram:
• This includes the “$A$ only” region, the “$B$ only” region, and the overlap.
• In other words, everything inside either circle.
• $A \cap B$ (intersection) is everything in both $A$ and $B$. On the diagram:
• This is only the overlapping region of the two circles.

By shading the appropriate regions, you can represent these expressions visually without writing them in symbols.

Showing complements and differences

A two-set Venn diagram can also show complements and differences:

• The complement $A^c$ (or sometimes $U \setminus A$) is everything in $U$ that is not in $A$.
• On the diagram: everything in the rectangle outside circle $A$.
• The set difference $A \setminus B$ means “in $A$ but not in $B$”.
• On the diagram: the “$A$ only” part of circle $A$ (the part not overlapping with $B$).

Similarly, $B \setminus A$ is the “$B$ only” region.

It is useful to practice identifying each of the four regions in terms of set expressions such as:

• $A \cap B$,
• $A \setminus B$,
• $B \setminus A$,
• $U \setminus (A \cup B)$.

Three-set Venn diagrams

With three sets $A$, $B$, and $C$, a Venn diagram uses three overlapping circles inside the universal set $U$. Each additional set creates more possible regions. For three sets, there are eight regions in total:

1. In none of the sets: outside all three circles (in $U$ but not in $A$, $B$, or $C$).
2. In $A$ only.
3. In $B$ only.
4. In $C$ only.
5. In $A$ and $B$ only (but not $C$).
6. In $A$ and $C$ only (but not $B$).
7. In $B$ and $C$ only (but not $A$).
8. In all three sets $A$, $B$, and $C$.

Each region can be described by a combination of “in” or “not in” for $A$, $B$, and $C$. For example:

• “In $A$ and $B$ only” corresponds to elements that are:
• in $A$,
• in $B$,
• not in $C$.
• “In all three sets” is the tiny central region where all three circles overlap; this is $A \cap B \cap C$.

Unions and intersections with three sets

Some common expressions and their regions:

• $A \cup B \cup C$: everything inside at least one of the circles.
• All regions except the one that is outside all three circles.
• $A \cap B$: all regions that are in both $A$ and $B$ (whether or not they are in $C$).
• This includes both:
• the “$A$ and $B$ only” region, and
• the central “$A$ and $B$ and $C$” region.
• $A \cap B \cap C$: only the central region, where all three circles overlap.
• $(A \cup B) \cap C$: in $C$ and in at least one of $A$ or $B$.
• On the diagram, this is the part of circle $C$ that lies inside either $A$ or $B$ (or both).

Practicing by shading these regions is a good way to connect symbolic expressions with the picture.

Differences and complements with three sets

You can also express differences and complements, though the regions can be smaller:

• $A \setminus (B \cup C)$: in $A$, but not in $B$ or $C$.
• On the diagram: the “$A$ only” region.
• $(A \cap B) \setminus C$: in both $A$ and $B$, but not in $C$.
• On the diagram: the “$A$ and $B$ only” region (excluding the central one that includes $C$).
• $(A \cup B)^c$: everything in $U$ that is in neither $A$ nor $B$.
• On the diagram: outside both $A$ and $B$ circles (this might include parts of $C$ if $C$ is drawn).

Working with these examples slowly, region by region, helps avoid confusion.

Using Venn diagrams with actual elements

So far we have talked about regions in general. You can also place specific elements inside the diagram:

• If $x \in A$ and $x \notin B$, write $x$ in the “$A$ only” region.
• If $y \in A$ and $y \in B$, write $y$ in the overlap $A \cap B$.
• If $z$ is in $U$ but not in $A$ or $B$, write $z$ outside the circles but inside the rectangle.

For three sets, you decide which of the eight regions each element belongs to, based on:

• whether it is in $A$ or not,
• whether it is in $B$ or not,
• whether it is in $C$ or not.

This is helpful for classification problems, or for organizing information into categories.

Counting with Venn diagrams

Venn diagrams are also useful for counting how many elements are in different parts of sets, especially with overlapping groups. Instead of writing elements explicitly, you write numbers in regions.

As a simple example with two sets:

• Suppose in a survey:
• 15 people like tea,
• 12 people like coffee,
• 5 people like both tea and coffee.

To fill a two-set Venn diagram:

1. Write 5 in the overlap $T \cap C$ (people who like both).
2. People who like tea only: $15 - 5 = 10$, so write 10 in the “tea only” region.
3. People who like coffee only: $12 - 5 = 7$, so write 7 in the “coffee only” region.

If you also know the total number of people surveyed, you can find how many are outside both sets by subtracting the total in all circles from the total number surveyed.

With three sets, a similar method works, but you must be careful to start from the region that belongs to the most sets (usually the central $A \cap B \cap C$ region) and work outward. The actual formula-based counting belongs to other chapters; here the focus is on placing the numbers correctly in the regions.

Representing logical situations

Because Venn diagrams deal with “in” and “not in” categories, they can also illustrate logical statements that involve “and”, “or”, and “not”.

• “$P$ and $Q$” corresponds to the intersection region where both sets $P$ and $Q$ overlap.
• “$P$ or $Q$” (inclusive “or”) corresponds to the union region, everything inside at least one of the sets.
• “not $P$” corresponds to the complement of $P$ (everything in $U$ outside the region for $P$).

In this way, Venn diagrams provide an intuitive picture for some logical ideas, especially when you want to visualize how conditions overlap.

Limitations of Venn diagrams

Venn diagrams are extremely helpful for:

• small numbers of sets (usually two or three),
• basic unions, intersections, and complements,
• simple counting questions,
• visualizing logical combinations.

However:

• They become hard to draw and read for many sets (four or more).
• They do not show every detail of a set (like order or structure), only membership.
• The picture is a guide; the formal meaning still comes from the set notation.

In this course, you can think of Venn diagrams as a bridge between verbal descriptions (“some people like both” or “in $A$ but not in $B$”) and symbolic set expressions like $A \cap B$, $A \cup B$, and $A \setminus B$.