Set Notation: Union, Intersection & Complement

Grade 5

The three key symbols

Set notation questions look intimidating until you know what three symbols mean:

A ∩ B — the intersection. This means "in A and B" — the overlap region only.

A ∪ B — the union. This means "in A or B (or both)" — everything inside either circle.

A′ — the complement. This means not in A — everything outside circle A, including the neither region.

Mapping symbols to regions

Every symbol maps directly to a shaded region of the Venn diagram. When you see a set notation expression, sketch the diagram and shade the region it describes — then count what's in the shading.

A ∩ B is just the lens. A ∪ B is the whole of both circles (the lens plus both side bits). A′ is everything outside circle A — right-hand bit of B plus the neither region.

Worked example

EXAMPLE

ξ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. A = {even numbers}. B = {multiples of 3}. Find (a) A ∩ B, (b) A ∪ B, (c) P(A′).

A = {2, 4, 6, 8, 10}. B = {3, 6, 9}. (a) A ∩ B = numbers in both = {6}. (b) A ∪ B = numbers in A or B or both = {2, 3, 4, 6, 8, 9, 10} — that's 7 numbers. (c) A′ = numbers not in A = {1, 3, 5, 7, 9} — that's 5 numbers. P(A′) = 5/10 = 1/2.

Nested notation

Expressions like (A ∪ B)′ mean you find A ∪ B first, then take the complement. Work from the inside out — don't apply the ′ to each part separately.

Try this one

For two sets \( A \) and \( B \), \( n(A) = 20 \), \( n(B) = 14 \) and \( n(A \cup B) = 27 \).

(a) Work out \( n(A \cap B) \). [2]

(b) Work out \( n(A \cap B') \), the number in \( A \) only. [2]

Nova's hint:

This uses the addition rule connecting the sizes of two sets, their union and their overlap.

Ask yourself: 'when I add \( n(A) \) and \( n(B) \), how many times is the overlap counted?' Rearranging the rule gives the size of the intersection.

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