Sets and elements
A set is a collection of distinct objects, listed inside curly brackets. Each object is an element of the set. We use capital letters for sets, so \(A = \{1, 2, 3, 4, 5\}\) means \(A\) is the set of whole numbers from \(1\) to \(5\).
- Element
- A member of a set. We write \(3 \in A\) to say "3 is an element of \(A\)", and \(7 \notin A\) to say "7 is not an element of \(A\)".
The universal set, written \(U\) (sometimes \(\xi\)), is the full collection of everything under discussion. If we are talking about a single ordinary die, then \(U = \{1, 2, 3, 4, 5, 6\}\), and every other set in the problem is drawn from inside it.
Union and intersection
Two operations combine sets. The union gathers everything in either set; the intersection keeps only what is in both.
A memory hook: \(\cup\) looks like a cup that scoops everything up (union, "or"), while \(\cap\) is a cap that only covers the overlap (intersection, "and").
Take \(A = \{1, 2, 3, 4, 5\}\) and \(B = \{2, 4, 6\}\) inside \(U = \{1, 2, 3, 4, 5, 6\}\):
Notice each element is written once in the union, even though \(2\) and \(4\) sit in both sets. Sets never repeat an element.
Or in maths means at least one
The union uses "or" in the mathematical sense: an element belongs to \(A \cup B\) if it is in \(A\), in \(B\), or in both. Everyday "or" often means one or the other but not both, so watch the difference.
Complement and the empty set
The complement of \(A\), written \(A'\), is everything in the universal set that is not in \(A\). It is the "everything else" set.
With \(U = \{1, 2, 3, 4, 5, 6\}\) and \(A = \{1, 2, 3, 4, 5\}\), the complement is \(A' = \{6\}\), because \(6\) is the only member of \(U\) left out of \(A\).
The empty set, written \(\varnothing\) or \(\{\ \}\), is the set with no elements at all. It turns up whenever two sets share nothing, for example \(A \cap B = \varnothing\) means \(A\) and \(B\) have no common element. Such sets are called disjoint.
Let \(U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}\), \(A = \{2, 4, 6, 8, 10\}\) (even numbers) and \(B = \{3, 6, 9\}\) (multiples of 3). Find \(A \cap B\), \(A \cup B\) and \(A'\).
The symbols at a glance
Keep this reference handy until the notation feels natural.
| Symbol | Name | Meaning |
|---|---|---|
| \(\in\) | is an element of | The object belongs to the set |
| \(\cup\) | union | In one set or the other, or both |
| \(\cap\) | intersection | In both sets at once |
| \(A'\) | complement | Everything in \(U\) not in \(A\) |
| \(\varnothing\) | empty set | A set with no elements |
| \(U\) | universal set | Everything under discussion |
Where this is assessed
Reading and writing the symbols correctly is Criterion A (Knowing and understanding). Laying sets out with clear brackets and no repeated elements is Criterion C (Communicating). Using sets to sort real data, such as survey responses, links to Criterion D (Applying in real-life contexts).
Check yourself
Answer each before revealing the solution.
1. If \(A = \{1, 3, 5, 7\}\) and \(B = \{2, 3, 5, 8\}\), what is \(A \cap B\)? +
The intersection keeps elements in both sets. \(3\) and \(5\) appear in each, so \(A \cap B = \) \(\{3, 5\}\).
2. With the same \(A\) and \(B\), write down \(A \cup B\). +
The union collects every element from either set, each listed once: \(A \cup B = \) \(\{1, 2, 3, 5, 7, 8\}\).
3. Let \(U = \{1, 2, 3, 4, 5, 6\}\) and \(A = \{1, 2, 3\}\). Find \(A'\), and state \(A \cap A'\). +
The complement is the members of \(U\) not in \(A\): \(A' = \{4, 5, 6\}\). A set and its complement share nothing, so \(A \cap A' = \) \(\varnothing\).
Part of the Standard Maths library. Spotted an error or want a topic added? That feedback makes the notes better.