MathematicsChapter 3: Sets and Relations
Three Sets and Venn Diagrams
With three sets A, B, C inside a universal set, a Venn diagram has seven inner regions plus the outside.
Definition
With three sets A, B, C inside a universal set, a Venn diagram has seven inner regions plus the outside. Start filling from the triple overlap A n B n C, then the pairwise-only regions, then only-A, then outside. Inclusion-exclusion: n(A u B u C) = n(A)+n(B)+n(C) - n(A n B)-n(B n C)-n(C n A) + n(A n B n C).
Overview
A word problem method:
The Seven Regions
Label them. 'A and B but not C' is the lens without the centre. 'Only A' is A minus both others. Students dump the given 'n(A n B)' into the pairwise lens without subtracting the centre — that double-counts.
Examples 20 in A and B includes those also in C · Draw first, then numbers · Check the total against the universal set
Counting
If 100 students, 40 in A, and the regions you filled already sum to 90, the outside is 10. A survey that sums over 100% of people in overlapping clubs is a clue you must use a Venn, not add the headlines.
Examples Clubs: cricket, football, hockey · Do not add 40+35+30 if there is overlap · Show the working on the diagram
In simple words
n(A u B) = n(A)+n(B)-n(A n B) is the two-set case. Three sets need the extra plus of the triple overlap after subtracting the three pairs.
Key points
- fill from the centre
- inclusion-exclusion
- not C still in A n B
- subtract the centre
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Three Sets and Venn Diagrams
Sticky Notes · Quick Revision
Core Idea
- With three sets A, B, C inside a universal set, a Venn diagram has seven inner regions plus the outside.
Key Points
- fill from the centre
- inclusion-exclusion
Simple Check
- Explain three sets and venn diagrams in your own words and connect it to one chapter example.
How to Answer
- Define the idea, show the main rule or evidence, apply it, and check the final statement.
Common Mistake
- n(A u B) = n(A)+n(B)-n(A n B) is the two-set case. Three sets need the extra plus of the triple overlap after subtracting the three pairs.
FBISE Exam Tip
- Use the exact term, show working where needed, and include units or labelled evidence.
60-second revision
- fill from the centre
- inclusion-exclusion
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