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Cambridge IGCSE Mathematics — 0580 Extended

Topic 1.2: Number — Sets & Venn Diagrams

Set notation

SymbolMeaningExample
\(\mathcal{E}\) or \(E\)Universal set — all elements under consideration\(\mathcal{E} = \{1,2,\ldots,10\}\)
\(\in\), \(\notin\)Is / is not an element of\(3 \in A\), \(7 \notin B\)
\(\emptyset\)Empty set (no elements)\(A \cap B = \emptyset\)
\(\subseteq\)Is a subset of (every element of \(A\) is in \(B\))\(\{2,4\} \subseteq \{2,4,6\}\)
\(A'\)Complement of \(A\) (in \(\mathcal{E}\), not in \(A\))If \(\mathcal{E}=\{1,2,3,4\}\), \(A=\{1,2\}\), then \(A'=\{3,4\}\)
\(A \cup B\)Union — in \(A\) or \(B\) or both\(\{1,2\}\cup\{2,3\}=\{1,2,3\}\)
\(A \cap B\)Intersection — in both \(A\) and \(B\)\(\{1,2\}\cap\{2,3\}=\{2\}\)
\(n(A)\)Number of elements in \(A\)\(n(\{5,5,7\})=2\) (no repeats)
Two-set Venn diagram inside universal set E showing regions A only, A intersect B, and B only
Diagram 1: Two-set Venn diagram — shade the region that matches the set expression in the question.

Reading Venn diagrams

Method

  1. Write \(\mathcal{E}\) and list every element shown (or given).
  2. Place elements in the correct region: \(A\) only, \(B\) only, \(A \cap B\), or outside both.
  3. For \(A'\): all elements in \(\mathcal{E}\) not in \(A\).
  4. Count carefully for \(n(A)\), \(n(A \cup B)\), or \(n(A \cap B)\).

\(\mathcal{E} = \{1,2,3,4,5,6,7,8\}\). \(A = \{2,4,6,8\}\), \(B = \{1,2,3,4\}\). Find \(A \cap B\) and \(n(A \cup B)\).

Venn diagram with elements placed in regions showing A intersect B equals two and four and union count six
Place each element in the Venn diagram — overlap gives A ∩ B, total in either set gives n(A ∪ B).

With the same sets, list the elements of \(A' \cap B\).

Shaded B-only region showing A prime intersect B equals one and three
A′ ∩ B is the part of B that lies outside A — elements {1, 3}.

Try this

\(\mathcal{E}=\{p,q,r,s,t\}\), \(A=\{p,q,r\}\), \(B=\{q,r,s\}\). Find \(n(A \cap B')\).

Show answer
Answer

  1. \(B' = \mathcal{E} - B\) — list \(\mathcal{E}\) and \(B\), then remove elements of \(B\)

    \[ \mathcal{E}=\{p,q,r,s,t\},\quad B=\{q,r,s\} \]
    \[ B' = \{p,\, t\} \]

  2. \(A \cap B'\) — elements in both \(A\) and \(B'\)

    \[ A=\{p,q,r\}, \quad A \cap B' = \{p\} \]

  3. Count the elements

    \[ n(A \cap B') = 1 \]

Three sets

For three sets \(A\), \(B\), \(C\), each region of the Venn diagram corresponds to a combination such as \(A \cap B \cap C'\) (in \(A\) and \(B\) but not \(C\)).

In a class of \(30\) students: \(18\) study French (\(F\)), \(15\) study Spanish (\(S\)), \(10\) study both. Find \(n(F \cup S)\) and \(n(F' \cap S)\).

French and Spanish Venn diagram with region counts eight, ten, and five showing union twenty three and F prime intersect S five
Fill the Venn regions first, then apply n(F ∪ S) = n(F) + n(S) − n(F ∩ S).

Exam Traps

  • \(A \cup B\) means or (include the overlap once) — do not add \(n(A)+n(B)\) without subtracting \(n(A \cap B)\).
  • \(A'\) is always relative to \(\mathcal{E}\) — state \(\mathcal{E}\) before finding a complement.

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