Set Operations Using Venn Diagrams

Good to see you again. In the previous lesson, you practised writing finite sets clearly and distinguishing membership, such as $3\in A$, from subsets, such as $\{3\}\subseteq A$. Those skills now let you work confidently with two sets at once.

This final revision lesson focuses on three operations that commonly appear in a sets test:

• union: elements in $A$ or $B$;
• intersection: elements in both $A$ and $B$;
• complement: elements in the universal set that are not in a given set.

You will also use Venn diagrams to show where every element belongs.

---

1. The universal set: the boundary of the question

Before finding a complement or filling a Venn diagram, identify the universal set. It contains every object allowed in that particular question.

It may be written as $U$, $E$, or sometimes $\xi$, depending on the textbook. For example:

$$U=\{1,2,3,4,5,6,7,8,9,10\}$$

A Venn diagram represents $U$ as a rectangle. Any set inside the question is drawn as a circle inside that rectangle.

Suppose:

$$A=\{2,4,6,8,10\}$$ $$B=\{2,3,5,7\}$$

Both $A$ and $B$ must be subsets of $U$. Notice that numbers such as 1 and 9 are in $U$, even though they are not in either $A$ or $B$. In a Venn diagram, they belong inside the rectangle but outside both circles.

This matters especially for complements: you only consider elements inside the stated universal set. You never include numbers from outside it.

---

2. Intersection: the overlap means “and”

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

$$A\cap B$$

Read $\cap$ as “intersection” or “and.”

Using the sets above:

$$A=\{2,4,6,8,10\}$$ $$B=\{2,3,5,7\}$$

The only number in both sets is 2:

$$A\cap B=\{2\}$$

In a Venn diagram, 2 goes in the overlapping middle region.

A reliable method is to inspect one set and ask whether each element is also in the other:

Element of $A$	Is it in $B$?	Put it in intersection?
2	Yes	Yes
4	No	No
6	No	No
8	No	No
10	No	No

The word and is the important clue:

• “even and prime” means intersection;
• “members of $A$ and $B$” means $A\cap B$.

If no elements are shared, the intersection is the empty set:

$$A\cap B=\varnothing$$

Sets with no elements in common are called disjoint sets.

---

3. Union: “or” includes both

The union of two sets contains every element that is in $A$, in $B$, or in both.

$$A\cup B$$

Read $\cup$ as “union” or “or.”

For the same sets:

$$A=\{2,4,6,8,10\}$$ $$B=\{2,3,5,7\}$$

Start by listing the elements of $A$, then add any new elements from $B$:

$$A\cup B=\{2,3,4,5,6,7,8,10\}$$

The number 2 appears in both sets, but it is written once only in the union. A set never repeats elements.

In mathematics, “or” means one or the other or both. So every element in the overlap is included in a union.

The image’s union diagram shades both circles, including their overlap. Its intersection diagram shades only the shared region. For this lesson, focus particularly on those two diagrams and the complement diagram.

---

4. Complement: “not in the set,” but still in the universal set

The complement of $A$ means all the elements of the universal set that are not in $A$.

It is often written as:

$$A^c$$

Some books write $A^{\prime }$ instead. Both mean “not $A$.” Use the notation your teacher or textbook uses.

Return to:

$$U=\{1,2,3,4,5,6,7,8,9,10\}$$ $$A=\{2,4,6,8,10\}$$

The elements in $U$ that are not in $A$ are 1, 3, 5, 7, and 9. Therefore:

$$A^c=\{1,3,5,7,9\}$$

On a Venn diagram, $A^c$ is everything in the rectangle outside circle $A$. If there is also a circle $B$, the complement of $A$ includes:

• the part of $B$ outside $A$, and
• the region outside both circles.

It does not include anything inside $A$, including the overlap.

This is why the universal set must always be stated. For instance, if instead

$$U=\{1,2,3,4,5,6\}$$

and

$$A=\{2,4,6\},$$

then:

$$A^c=\{1,3,5\}$$

The complement is not “all odd numbers in existence.” It is only the odd numbers that lie in the particular universal set.

---

5. Filling in a Venn diagram accurately

When a question gives you a universal set and two subsets, use the same order every time. It prevents the most common mistake: putting an element in one circle when it really belongs in the overlap.

Consider:

$$U=\{1,2,3,4,5,6,7,8,9,10,11,12\}$$ $$P=\{2,3,5,7,11\}$$ $$Q=\{1,2,3,4,6,12\}$$

Step 1: Fill the intersection first

Find the values in both sets:

$$P\cap Q=\{2,3\}$$

Put 2 and 3 in the overlap.

Step 2: Fill the $P$-only region

Remove the shared elements from $P$:

$$\{5,7,11\}$$

These go in circle $P$, but not in the overlap.

Step 3: Fill the $Q$-only region

Remove the shared elements from $Q$:

$$\{1,4,6,12\}$$

These go in circle $Q$, but not in the overlap.

Step 4: Check the universal set for leftovers

The elements of $U$ not placed anywhere yet are:

$$\{8,9,10\}$$

Put these inside the rectangle but outside both circles.

Your completed diagram has four regions:

Region	Elements
$P$ only	$\{5,7,11\}$
$P\cap Q$	$\{2,3\}$
$Q$ only	$\{1,4,6,12\}$
Neither $P$ nor $Q$	$\{8,9,10\}$

From that one diagram, you can read off several answers:

$$P\cap Q=\{2,3\}$$ $$P\cup Q=\{1,2,3,4,5,6,7,11,12\}$$ $$P^c=\{1,4,6,8,9,10,12\}$$

Notice that $P^c$ includes both the $Q$-only section and the “outside both” section. It includes every element of $U$ outside circle $P$.

---

6. Reading answers directly from a completed diagram

A Venn diagram is not just a picture; it is a way to sort every element of the universal set into exactly one region.

Suppose a finished diagram has:

Region	Elements
$A$ only	$\{1,5\}$
$A\cap B$	$\{3,8\}$
$B$ only	$\{2,7,10\}$
Outside both sets	$\{4,6,9\}$

You can answer by selecting regions carefully.

Finding $A$

Everything inside circle $A$:

$$A=\{1,3,5,8\}$$

Finding $B$

Everything inside circle $B$:

$$B=\{2,3,7,8,10\}$$

Finding the intersection

Only the overlap:

$$A\cap B=\{3,8\}$$

Finding the union

Everything in either circle:

$$A\cup B=\{1,2,3,5,7,8,10\}$$

Finding the complement of $A$

Everything outside circle $A$, but still inside the universal rectangle:

$$A^c=\{2,4,6,7,9,10\}$$

A quick visual memory aid is:

Operation	Meaning	Region to use
$A\cap B$	$A$ and $B$	Overlap only
$A\cup B$	$A$ or $B$, including both	Both circles
$A^c$	Not in $A$	Everything outside $A$, within $U$

---

7. Test-day checklist

When you meet a union, intersection, complement, or Venn-diagram question on Tuesday, pause for these checks:

1. Write down the universal set. Every listed element should come from it.
2. For an intersection, look for elements in both sets.
3. For a union, include elements from either set, but list each one once.
4. For a complement, look only inside the universal set and collect what is outside the named circle.
5. When drawing a diagram, place the overlap first.
6. Finally, check that every element of the universal set appears somewhere in the rectangle, exactly once.

A useful five-minute revision activity is to redraw one two-circle Venn diagram from your textbook without looking at the solution. Afterward, check: overlap first, separate regions next, leftover universal-set elements last.

---

Wrap-up

The three main operations are linked to simple words:

$$A\cap B\text{means “in A and B”}$$ $$A\cup B\text{means “in A or B, including both”}$$ $$A^c\text{means “in the universal set but not in A”}$$

The rectangle in a Venn diagram is essential because it represents the universal set. The overlap gives the intersection, both circles together give the union, and everything outside a circle gives that set’s complement.

You have now revised both main parts of your sets test: set notation and relationships, followed by operations and Venn diagrams.