ZIMSEC O Level · Mathematics · Form 1, Form 2, Form 3
A set is a well-defined collection of objects called elements.
Notation
- ∈ means “is an element of”; ∉ means “is not an element of”.
- n(A) is the number of elements in set A.
- ξ is the universal set (everything being considered).
- A ∪ B (union) contains elements in A or B or both. A ∩ B (intersection) contains elements in both.
- A′ (complement) contains elements of ξ that are not in A.
- ∅ or { } is the empty set. A set with n elements has 2ⁿ subsets.
Two-set problems
Start with the intersection, then fill in the rest of each set, then the outside region.
Worked example: In a class of 40, 25 study Shona, 20 study Ndebele and 8 study neither. How many study both?
Learners studying at least one = 40 − 8 = 32
n(S ∪ N) = n(S) + n(N) − n(S ∩ N) → 32 = 25 + 20 − x
x = 13 learners study both.
Three sets (Form 3)
Use three overlapping circles. Fill the centre (all three) first, then the pairs, then each set alone. Write an equation for the total.
Exam tip: Read every region carefully: “only A” means A but not B. Many errors come from counting the overlap twice.
Check yourself
- ξ = {1, 2, …, 10}, A = {even numbers}, B = {multiples of 3}. List A ∩ B.
- How many subsets does {a, b, c, d} have?
- n(ξ) = 50, n(P) = 30, n(Q) = 26, n(P ∩ Q) = 12. Find n((P ∪ Q)′).
Show answers
- {6}
- 2⁴ = 16
- 50 − (30 + 26 − 12) = 6
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