Sets, Relations and Functions
What the exam tests
This domain checks fluent use of set notation and operations, counting with Venn diagrams, the definition of a relation as a subset of a Cartesian product, and the function idea of one output per input. Questions often combine a small Venn-diagram count with a domain/range or composite-function item.
Core rules
- For finite sets, (n(A\cup B)=n(A)+n(B)-n(A\cap B)); for three sets subtract the three pairwise intersections and add back the triple intersection.
- De Morgan: ((A\cup B)'=A'\cap B') and ((A\cap B)'=A'\cup B').
- A relation (R) from (A) to (B) is any subset of (A\times B). Its domain is the set of first components; its range is the set of second components; the codomain is (B).
- A function assigns exactly one image to each domain element. It is one-one if distinct inputs have distinct images, onto if range equals codomain, and bijective if both. An inverse exists only for a bijection.
Worked example
Let (A={1,2,3}), (B={2,4,6,8}) and (R={(1,2),(2,4),(3,6)}). Each element of (A) appears once, so (R) is a function. Domain (={1,2,3}), range (={2,4,6}), codomain (=B). Since 8 is in the codomain but not the range, the function is not onto. It is one-one because the three images are different.
Traps
Do not confuse (\varnothing) with ({0}); the first has no elements. A relation need not be a function. Range and codomain are not interchangeable. When finding a complement, state the universal set. Check whether the inverse is being asked for a relation or a function.