Sets, Relations and Functions
Sets, Relations and Functions introduces the language of collections, ordered pairs, mappings, domain and range. Questions are definition-heavy and reward precise notation.
Core rules
- A set is a well-defined collection; order and repetition do not change it. The empty set ∅ has no elements.
- Union A ∪ B contains elements in A or B; intersection A ∩ B contains elements in both; difference A − B contains elements in A but not B.
- n(A ∪ B) = n(A) + n(B) − n(A ∩ B).
- A relation from A to B is a subset of A × B. A function assigns exactly one output to each input.
- Domain = set of allowed inputs; range = set of actual outputs. For f(x) = 2x + 1, f(3) means substitute x = 3.
Worked example
Let A = {1, 2, 3} and B = {3, 4}. Then A ∪ B = {1, 2, 3, 4}, A ∩ B = {3}, and A − B = {1, 2}. Also n(A ∪ B) = 3 + 2 − 1 = 4, confirming the union has four elements.
Traps
{3} is a set, not the number 3; do not mix the element symbol ∈ with the subset symbol ⊆. A relation such as {(1, 2), (1, 3)} is not a function because the input 1 has two outputs. In a function, every element of the domain must be used, but not every element of the codomain need appear in the range.