Describing collections with sets and counting with Venn diagrams
A set is a well-defined collection of objects, written in roster form, such as {2, 4, 6}, or in set-builder form, such as {x : x is an even natural number less than 8}. A is a subset of B when every element of A is also an element of B, and the empty set is a subset of every set. A set with n elements has 2ⁿ subsets, which together form its power set. The universal set U contains all the elements under discussion.
The union A ∪ B contains the elements in A or B or both, the intersection A ∩ B contains the elements common to both, the difference A − B contains the elements in A but not in B, and the complement A′ contains the elements of U that are not in A. De Morgan's laws state that (A ∪ B)′ = A′ ∩ B′ and (A ∩ B)′ = A′ ∪ B′. Venn diagrams show these operations as overlapping circles inside a rectangle. For counting, n(A ∪ B) = n(A) + n(B) − n(A ∩ B): if 25 students play cricket, 18 play football and 10 play both, then 33 play at least one of the two games.
Why it matters
Sets give mathematics a precise language that is used for relations, functions, probability and logic. Venn diagrams are a practical way to sort survey results, such as how many people use two different apps, and the same reasoning underlies database searches that combine conditions with AND, OR and NOT. Set questions are common in Class 11 exams and in the reasoning sections of competitive tests.
Key terms
- Subset
- A set whose every element is also an element of another given set.
- Power set
- The set of all subsets of a given set, including the empty set and the set itself.
- Complement
- All elements of the universal set that do not belong to the given set.
- De Morgan's laws
- Rules stating that the complement of a union is the intersection of complements, and the reverse.