Revision notes
Describe sets in words and symbols, count their elements and subsets, and find unions, intersections and complements.
Aisha's basket has mangoes, oranges, pawpaw and guava. The fruits she likes form one group; the fruits that are in season form another. Mathematicians call such groups sets, and they have a neat shorthand for talking about them.
A set is a well-defined collection of objects called elements. We list them in curly brackets: A = {2, 3, 5, 7}.
We can also describe a set with a rule: A = {x : x is a prime number, x < 10} is read 'the set of x such that x is a prime number less than 10'.
n(A) is the number of elements in A. Here n(A) = 4. The empty set ∅ = { } has no elements, and the universal set ξ contains everything being considered.
🔑 Key idea: ∈ means 'is an element of': 3 ∈ A, but 4 ∉ A.
B ⊂ A means every element of B is also in A. If A = {1, 2, 3} then {1, 3} ⊂ A.
Each element is either in a subset or out of it — 2 choices per element. So a set with n elements has 2ⁿ subsets. {a, b, c} has 23 = 8 subsets, including ∅ and the set itself.
A proper subset is any subset except the set itself, so there are 2ⁿ − 1 of them.
🔑 Key idea: A set with n elements has 2ⁿ subsets.
A ∪ B (union) is everything in A or B or both — list each element once. A ∩ B (intersection) is only what is in both.
A′ (complement) is everything in ξ that is NOT in A.
If ξ = {1, …, 8}, A = {1, 2, 3, 4} and B = {3, 4, 5}: A ∪ B = {1, 2, 3, 4, 5}, A ∩ B = {3, 4} and A′ = {5, 6, 7, 8}.
🔑 Key idea: ∪ = in either (join), ∩ = in both (overlap), ′ = not in.
A = {x : x is an odd number, 7 < x < 21}. List the elements of A, then give n(A).
Step 1: List the elements
A = {9, 11, 13, 15, 17, 19}
Step 2: Count them
n(A) = 6
Answer: 6
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