Sets and set notation

A set is a collection of things, called its elements or members, written inside curly brackets. $A = \{2, 4, 6, 8\}$ is a set with four elements. $6 \in A$ means “6 is an element of $A$”, and $5 \notin A$ means “5 is not”.

The order doesn’t matter, and an element is written only once: $\{2, 4, 6, 8\}$ and $\{8, 6, 4, 2\}$ are the same set.

Describing a set

There are two ways to write a set down.

• List the elements: $B = \{11, 13, 17, 19\}$.
• Describe them with a rule (set-builder notation): $B = \{x : x \text{ is a prime number}, 10 < x < 20\}$. Read the colon as “such that”.

$n(A)$ means the number of elements in $A$. For $B$ above, $n(B) = 4$.

When the rule is an inequality and ”$x$ is an integer”, list the whole numbers the inequality allows. Watch whether each end is $<$ (left out) or $\le$ (kept in).

The universal set, the empty set and subsets

The universal set, $U$ (WAEC often writes $\mu$ or $\xi$), holds every element being talked about in the question. It’s drawn as the rectangle round a Venn diagram.

The empty set, $\varnothing$ or $\{\,\}$, has no elements. $\{x : x \text{ is a whole number}, 2 < x < 3\}$ is empty, because there is no whole number between 2 and 3.

$B$ is a subset of $A$, written $B \subseteq A$, when every element of $B$ is also in $A$. Two sets with no elements in common are disjoint.

A set with $n$ elements has $2^n$ subsets, counting $\varnothing$ and the set itself. $\{a, b, c\}$ has $2^3 = 8$: $\varnothing$, $\{a\}$, $\{b\}$, $\{c\}$, $\{a, b\}$, $\{a, c\}$, $\{b, c\}$, $\{a, b, c\}$.

Union, intersection and complement

These are the three operations that come up in almost every sets question. The shaded region shows each one.

• $A \cup B$, the union: every element in $A$ or $B$ or both. Each element is written only once.
• $A \cap B$, the intersection: only the elements in both.
• $A'$, the complement of $A$: every element of $U$ that is not in $A$. So the complement depends on $U$.
• $A - B$ (sometimes written $A \setminus B$): the elements of $A$ that are not in $B$. It is the same as $A \cap B'$, the last diagram above.

Try it

Each element is written in its region, so to list a set, shade its regions and read the numbers in them. Switch on Your turn to shade the regions yourself before you check.

Two useful rules, called De Morgan’s laws, come from the diagrams:

“Not in either” is the same as “not in $A$ and not in $B$”. WAEC sometimes asks you to show one of them for given sets: work out each side separately and show they are the same set.

Three sets

With three sets, the circles make eight regions. Name each region by which circles it is inside and which it is outside.

Your turn