Basic Ideas
A set is a well-defined collection of objects. The objects are called elements, and the number of elements of A is written n of A. A is a subset of B if every element of A is in B. A proper subset is a subset that is not equal to the whole set. The empty set has no elements and is a subset of every set. The universal set contains all elements under discussion. The complement of A is the set of elements of U not in A. The union of A and B holds elements in either. The intersection holds elements in both. The difference A minus B holds elements in A but not in B. Two sets are disjoint if their intersection is empty.
Key Formulas
A set of n elements has 2 to the n subsets and 2 to the n minus 1 proper subsets. The number of elements in A union B equals n of A plus n of B minus n of A intersection B. For three sets, add the singles, subtract the pairwise overlaps and add the triple overlap. De Morgan's laws say the complement of a union is the intersection of the complements, and the complement of an intersection is the union of the complements. The number of elements in the Cartesian product is the product of the two counts.
Solving Venn Diagram Problems
Start with the innermost region, which is the triple overlap. Then find the pairwise-only regions by subtracting it. Then find the single-only regions. Finally, check that the total matches.
Shortcuts
For two sets, the number in at least one equals n of A plus n of B minus n of the overlap, and the number in neither is the total minus that. A set with 3 elements has 8 subsets, with 4 elements has 16 and with 5 elements has 32. Watch for the word "only".
Worked Examples
Example 1: If n of A is 30, n of B is 25, and n of A intersection B is 10, find n of A union B.
Solution: 30 plus 25 minus 10 equals 45.
Example 2: How many subsets does a 4-element set have?
Solution: 2 to the 4, which is 16. Of these, 15 are proper subsets.
Example 3: Of 50 students, 30 like cricket, 25 like football and 10 like both. How many like neither?
Solution: At least one is 45, so neither is 5.
Quick Check
Question 1: How many subsets does an n-element set have?
Answer: 2 to the n.
Question 2: State De Morgan's first law.
Answer: The complement of A union B equals A complement intersect B complement.
Question 3: What is n of A union B?
Answer: n of A plus n of B minus n of A intersection B.