Union ∪ joins two sets, intersection ∩ keeps only what they share, and the complement ′ lists what is outside a set but inside the universal set ξ.
This lesson starts SPM Mathematics set operations. It prepares you for two-set Venn diagram questions.
One universal set, five operations
Here is an original setup. Let ξ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}, A = {multiples of 3} and B = {even numbers}.
List the sets first: A = {3, 6, 9, 12} and B = {2, 4, 6, 8, 10, 12}. Then each operation is a matter of careful listing.
| Notation | Meaning | Elements |
|---|---|---|
| A ∪ B | In A or B or both | {2, 3, 4, 6, 8, 9, 10, 12} |
| A ∩ B | In both A and B | {6, 12} |
| A′ | In ξ but not in A | {1, 2, 4, 5, 7, 8, 10, 11} |
| (A ∪ B)′ | Outside both sets | {1, 5, 7, 11} |
| A′ ∩ B | Not in A, but in B | {2, 4, 8, 10} |
Check the union count: n(A) + n(B) − n(A ∩ B) = 4 + 6 − 2 = 8, and the union has 8 elements.
Where does the complement act?
The prime mark acts on the set it is attached to. In A′ ∩ B, only A is complemented, then the result is intersected with B.
In (A ∩ B)′, the brackets mean you find A ∩ B first and complement the whole result. For the sets above, A ∩ B = {6, 12}, so (A ∩ B)′ = {1, 2, 3, 4, 5, 7, 8, 9, 10, 11}.
The mistake that costs marks
The usual slip is treating A′ ∩ B and (A ∩ B)′ as the same. One gives {2, 4, 8, 10}, four elements, and the other gives ten elements.
| Step | Wrong | Right |
|---|---|---|
| Reading A′ ∩ B | Complement of the intersection | Complement A, then intersect with B |
| Result | {1, 2, 3, 4, 5, 7, 8, 9, 10, 11} | {2, 4, 8, 10} |
Another slip is listing A′ without ξ. If the question gives no universal set, you cannot list the complement, so look for it in the question stem.
A method for any expression
Work in layers. First list ξ, A and B. Then deal with anything in brackets, then complements, then ∩ or ∪ last.
Write each result as a set with curly brackets. Finally, cross-check the size against a counting formula when the operation allows one.
Check yourself
Let ξ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, P = {prime numbers} and Q = {odd numbers}. List P ∩ Q, P ∪ Q, (P ∪ Q)′ and P′ ∩ Q.
Answer
P = {2, 3, 5, 7} and Q = {1, 3, 5, 7, 9}.
P ∩ Q = {3, 5, 7}. P ∪ Q = {1, 2, 3, 5, 7, 9}. (P ∪ Q)′ = {4, 6, 8, 10}.
P′ = {1, 4, 6, 8, 9, 10}, and the elements of P′ that are also odd are P′ ∩ Q = {1, 9}. Count check: 4 + 5 − 3 = 6, which matches the union.
What to study next
Notation becomes useful when it labels a Venn diagram. Continue with solving two-set Venn diagram questions, and practise turning sentences into symbols in translating word statements into set notation.
If you would like a teacher to read symbols with you on your own questions, see online one-to-one Mathematics tuition.