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C1.2 Sets

Syllabus
0580–2028–2029
Topic
C1.2
Level
Core

Learning objectives

C1.2.1Understand and use set language, notation and Venn diagrams to describe• Understand and use set language, notation and Venn diagrams to describe sets. Venn diagrams are limited to two sets. The following set notation will be used: • n(A) Number of elements in set A • A′ Complement of set A • Universal set • A ∪ B Union of A and B • A ∩ B Intersection of A and B. Example definition of sets: A = {x: x is a natural number} B = {a, b, c, … } C = {x: a ⩽ x ⩽ b}.

Describe and count two sets with Venn diagrams

A set is a collection whose members are fixed by a rule or a list. A two-set Venn diagram sorts every member of the universal set into four regions: in both sets, in only one of them, or in neither.

Notation Meaning Region in a two-set diagram
n(A)n(A) number of elements in AA count all of circle AA
A′A' complement of AA everything in the universal set but outside AA
A∪BA\cup B union: in AA or BB or both both circles, including the overlap
A∩BA\cap B intersection: in both AA and BB the overlap only

The rectangle represents the universal set, written E\mathscr{E}. A complement is meaningful only after this universe is fixed: changing E\mathscr{E} can change A′A' even when AA stays the same.

Form What it says Example
list or roster write the elements between braces B={a,b,c,…}B=\{a,b,c,\ldots\}
set-builder name a variable and give its rule A={x:x is a natural number}A=\{x:x\text{ is a natural number}\}
bounded set-builder give the allowed range C={x:a≤x≤b}C=\{x:a\le x\le b\}; both endpoints are included

Place the intersection first. Then place members of AA that are not in BB in the AA-only region, members of BB that are not in AA in the BB-only region, and all remaining members of E\mathscr{E} outside both circles.

n(A∪B)=n(A)+n(B)−n(A∩B)n(A\cup B)=n(A)+n(B)-n(A\cap B)

The intersection is subtracted once because it was counted once in n(A)n(A) and again in n(B)n(B). Also, n(A′)=n(E)−n(A)n(A')=n(\mathscr{E})-n(A) when every element is counted within the same universal set.

In set language, ‘or’ is inclusive: an element in both sets belongs to A∪BA\cup B. Do not omit the overlap, and do not treat A′A' as everything imaginable outside AA—it means outside AA but still inside the stated universal set. This Core objective uses no more than two sets.