Set notation and Venn diagrams

Lesson overview

Statistics — sets and probability

A **set** is a collection of distinct objects. We use specific notation to describe relationships between sets: - $a \in A$ means ' $a$ is an **element** of $A$ ' - $A \cup B$ means ' $A$ **union** $B$ ' — everything in $A$ or $B$ or both - $A \cap B$ means ' $A$ **intersection** $B$ ' — only what is in both $A$ **and** $B$ - $A'$ (or $A^c$ ) means the **complement** of $A$ — everything NOT in $A$ - $\emptyset$ is the **empty set** - $A \subset B$ means ' $A$ is a **subset** of $B$ ' **Venn diagrams** represent sets visually: a rectangle (the universal set $\xi$ ) containing overlapping circles. The overlap shows $A \cap B$ , and the total covered area shows $A \cup B$ . For probability: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$

Understanding Venn diagram regions

In a Venn diagram with sets $A$ and $B$ : the overlap is $A \cap B$ , the total covered area is $A \cup B$ , and everything outside $A$ is $A'$ .

A \cap B

both

intersection

A \cup B

either or both

union

A'

not in A

complement

\emptyset

nothing

empty set

Intersection = AND, Union = OR, Complement = NOT.

Using set notation for probability

If $P(A) = 0.6$ , $P(B) = 0.4$ , and $P(A \cap B) = 0.2$ , then $P(A \cup B) = 0.6 + 0.4 - 0.2 = 0.8$ .

We subtract the intersection so we don't double-count.

Complement rule

The probability of NOT $A$ is $P(A') = 1 - P(A)$ . If $P(A) = 0.3$ , then $P(A') = 0.7$ .

$P(A) + P(A') = 1$ always.

Worked example

In a class of $30$ students, $18$ study French ( $F$ ) and $14$ study Spanish ( $S$ ). $6$ study both. A student is chosen at random. Find the probability they study French or Spanish.

$n(F) = 18$ , $n(S) = 14$ , $n(F \cap S) = 6$ , total $= 30$ .
Using the formula: $n(F \cup S) = n(F) + n(S) - n(F \cap S)$ .
$n(F \cup S) = 18 + 14 - 6 = 26$ .
$P(F \cup S) = \dfrac{26}{30} = \dfrac{13}{15}$ .

Try it

Identify what each symbol means, then use the appropriate formula.

Question 1

If $A = \{1, 2, 3, 4\}$ and $B = \{3, 4, 5, 6\}$ , what is $A \cap B$ ?

Question 2

If $P(A) = 0.7$ , what is $P(A')$ ?

Question 3

If $P(A) = 0.5$ , $P(B) = 0.4$ , and $P(A \cap B) = 0.2$ , what is $P(A \cup B)$ ?

Question 4

What does the symbol $\emptyset$ represent?

Question 5

If $A \subset B$ , which statement is true?

Common mistakes

Watch for these when working through the lesson.

Adding $P(A) + P(B)$ without subtracting $P(A \cap B)$ — this double-counts the overlap.
Confusing $\cup$ (union, OR) with $\cap$ (intersection, AND).
Forgetting that $P(A') = 1 - P(A)$ is often the quickest route to an answer.

Set notation and Venn diagrams

Lesson overview

Worked example

Try it

Common mistakes

Related topics

Independent practice will plug in here