Set Theory / Venn Diagrams

Set theory is used to group objects with common properties. Venn diagrams are the visual representation of these groups.

Set: A well-defined collection of distinct objects, called elements.
Key Operations:
- Union (A ∪ B): All elements that are in set A, or in set B, or in both.
- Intersection (A ∩ B): All elements that are in both set A and set B.
- Complement (A'): All elements in the universal set that are NOT in set A.
- Difference (A - B): All elements that are in set A but NOT in set B.

Cardinality Rules (Number of elements):
- For two sets: \( n(A \cup B) = n(A) + n(B) - n(A \cap B) \)
- For three sets: \( n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(B \cap C) - n(C \cap A) + n(A \cap B \cap C) \)
Understanding Venn Diagram Regions:
- 'Only A' corresponds to the region of A that does not intersect with any other set.
- 'At least two' means the sum of the intersections of two sets at a time plus the intersection of all three.
- 'Exactly one' means the sum of the 'only A', 'only B', and 'only C' regions.

Always Draw a Diagram: For any problem involving two or more overlapping categories, the first step should be to draw a Venn diagram. It provides immense clarity.
Start from the Inside Out: When filling values into the diagram, always begin with the most specific information, which is usually the central region (the intersection of all sets), and then work your way outwards.
Use Variables: If the central value isn't given, label it 'x' and express the other regions in terms of 'x'.

Pay extremely close attention to the language. Words like 'only', 'at least', 'at most', 'exactly', and 'none' have precise meanings. Misinterpreting one word can lead to a completely wrong answer.
Using the formula for three sets can be error-prone. Filling a Venn diagram is often a safer and more intuitive method.
Label all regions of your diagram clearly to avoid confusion during calculation.