Venn Diagrams and Set Operation
Venn Diagrams: Intersections and Unions
Intersections tend to be smaller because they require membership in all involved sets: a element must be in every set to be in the intersection. In contrast, unions represent any element that is in at least one of the sets.
Venn diagrams help visualize relationships between sets and can be used with surveys to answer questions like who likes certain items or activities.
Common notation:
- Sets: C = comedy, D = drama, S = science fiction.
- Intersections: C ∩ D, C ∩ S, D ∩ S.
- Triple intersection: C ∩ D ∩ S.
- Union: C ∪ D ∪ S.
Key principle: Inclusion-Exclusion for three sets:
- General formula:
|C \,\cup\, D \,\cup\, S| = |C| + |D| + |S| \ - |C \cap D| - |C \cap S| - |D \cap S| \ + |C \cap D \cap S|. - To find only-one-set regions (e.g., C only), use:
|C \,\text{only}| = |C| - |C \cap D| - |C \cap S| + |C \cap D \cap S|. - Note why the triple intersection is added back: it was subtracted twice when removing the pairwise overlaps, so we add it once to correct.
- General formula:
Worked example: Comedy/Drama/Sci-Fi problem
Given:
- (|C| = 26) (people who like comedy)
- (|C \cap D| = 7) (like comedy and drama)
- (|C \cap D \cap S| = 6) (like comedy, drama, and science fiction)
- (|C \cap S| = 4) (like comedy and science fiction)
Goal: find the number who like only comedy (C only).
Step 1: apply the C-only formula:
$$|C \text{\,only}| = |C| - |C \cap D| - |C