A7LAM CSCA ACADEMYCSCA Academy / Mathematics / Sets and inequalities
CSCA Mathematics · Lesson 2 of 16Sets and set operations
Ensembles et opérations · المجموعات والعمليات عليها
▶ Watch the video lesson (A7LAM students)
A set is a collection of distinct objects. CSCA questions ask you to combine sets and read set notation quickly.
Key ideas
- x ∈ A means x belongs to A; A ⊆ B means every element of A is in B.
- Union A ∪ B: elements in A or B. Intersection A ∩ B: elements in both.
- Complement ∁ᵤA: the elements of the universal set U that are not in A.
- A set with n elements has 2ⁿ subsets.
Formulas to know
- n(A ∪ B) = n(A) + n(B) − n(A ∩ B)
- ∁ᵤ(A ∪ B) = ∁ᵤA ∩ ∁ᵤB
- ∅ ⊆ A for every set A
Worked example
A = {1, 2, 3, 4} and B = {3, 4, 5, 6}. Find A ∩ B and A ∪ B.
- Both: 3 and 4 are in A and in B
- Either: list every element once: 1, 2, 3, 4, 5, 6
- Write: A ∩ B = {3, 4}, A ∪ B = {1, …, 6}
Answer: A ∩ B = {3, 4}; A ∪ B = {1, 2, 3, 4, 5, 6}
Practice questions
1. How many subsets does {a, b, c} have?
3 · 6 · 8 ✓ · 9
2³ = 8, including ∅ and the set itself.
2. U = {1, 2, 3, 4, 5, 6} and A = {2, 4, 6}. ∁ᵤA = ?
{1, 3, 5} ✓ · {2, 4, 6} · {1, 2, 3} · ∅
The elements of U that are not in A.
3. 20 students study French, 15 study Chinese and 5 study both. How many study at least one?
35 · 30 ✓ · 25 · 40
20 + 15 − 5 = 30.