AQA GCSE Mathematics Paper 1 (Higher), June 2025: Question 16

3 marks · Medium difficulty · Short Answer

Match set notation expressions involving events A and B to the corresponding shaded areas on Venn diagrams.

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Question

The question asks to match four set notation boxes on the left to five Venn diagrams with two overlapping circles, A and B, on the right. One is completed: 'A ∩ B' is joined to the second diagram, which has only the intersection shaded. The other boxes to match are 'A ∪ B', 'A' ∪ B', and 'A' ∩ B''. The available diagrams show: the entire union shaded; only the intersection shaded; only the area outside both circles shaded; the area outside A (including B and the background) shaded; and the symmetric difference (A only and B only) shaded.
Question text

16 A and B are two events.

Match each box on the left to the correct shaded area on the Venn diagrams.

One has been done for you.

[3 marks]

Mark scheme

Show the mark scheme The mark scheme shows matching lines: 'A ∩ B' is connected to the Venn diagram with only the intersection of A and B shaded; 'A ∪ B' is connected to the diagram with both circles entirely shaded; 'A' ∪ B' is connected to the diagram with B and the region outside both circles shaded; and 'A' ∩ B'' is connected to the diagram with only the region outside both circles shaded. Marks awarded: B3 for all three correct, B2 for two correct, B1 for one correct.

Q Answer Mark Comments

All three correct B2 two correct

B3

B1 one correct

Additional Guidance

B3

Two lines from a left-hand box is choice

How to answer it

Matching Set Notation to Venn Diagrams

AQA GCSE Mathematics • Probability & Sets • 3 Marks

What this question tests

Understanding and interpreting standard set theory notation (union ∪, intersection ∩, and complement ') and correctly linking each algebraic expression to its corresponding shaded region on a two-set Venn diagram within universal set ξ.

Question Walkthrough & Matches

Question 16 (Total: 3 marks)

✅ Correct Matches

Set Notation Target Venn Diagram (from top to bottom) Description of Shaded Area
A ∩ B 2nd diagram (Given as example) Only the central overlap region where circles A and B intersect.
A ∪ B 1st diagram (top) All of circle A, all of circle B, and the overlap (both complete circles shaded).
A' ∪ B 4th diagram Everything outside circle A PLUS all of circle B (only "A only" remains unshaded).
A' ∩ B' 3rd diagram The region outside both circles (neither in A nor in B).

Note: The 5th diagram shows the symmetric difference (only A and only B, leaving the overlap and exterior unshaded); it has no match.

Mark Scheme Breakdown:
• B3: All 3 lines correctly matched.
• B2: Any 2 lines correctly matched.
• B1: Any 1 line correctly matched.

💡 Key Knowledge: The 3 Core Symbols

  • Intersection ( ∩ ): Means "AND". Shading must satisfy both conditions simultaneously (overlap).
  • Union ( ∪ ): Means "OR" (or both). Shading includes anything belonging to either set.
  • Complement ( ' ): Means "NOT". For example, A' means everything outside circle A.
  • De Morgan's Law link: A' ∩ B' is identical to (A ∪ B)' — "NOT (A or B)", meaning completely outside both circles.

📐 Step-by-Step Logic for Tricky Sets

Decoding A' ∪ B :

  1. Identify A' : shade everything outside circle A (includes background + "B only").
  2. Identify B : shade all of circle B (includes "B only" + "overlap").
  3. Combine them ( ∪ ): combine both shaded areas. The only section left completely unshaded is "A only".
  4. This directly matches the 4th diagram.

🧠 Exam Technique & Strategy

  • Elimination: Count the diagrams vs. the labels. There are 4 boxes on the left (1 pre-completed) and 5 diagrams on the right, so exactly one diagram must remain unmatched.
  • Single line rule: Always draw exactly one clear straight line from each left box. The mark scheme states: "Two lines from a left-hand box is choice" — this awards zero marks for that item.
  • Use words first: Translate symbols into English before matching:
    • A ∪ B → "in A, in B, or both"
    • A' ∩ B' → "not in A AND not in B"

❌ Common Errors & Pitfalls

  • Confusing ∪ and ∩: Mixing up the union ("cup" collects all) and intersection ("cap" catches only overlap).
  • Misreading A' ∩ B' as A' ∪ B' :
    A' ∩ B' is only the outside region (3rd diagram).
    A' ∪ B' would be everything except the central intersection.
  • Failing to identify A' ∪ B : Students often overlook that the intersection A ∩ B gets shaded because it is inside B .
  • Messy drawings: Crossing out lines unclearly. If you make a mistake, rub it out fully or clearly put a cross through the unwanted line.

Topics

Probability · 3.5 Probability

Question and mark scheme from the AQA GCSE Mathematics examination, Paper 1 (Higher), June 2025. QuestionVault is an independent revision resource; questions remain the copyright of the awarding body.