STEP-BY-STEP LESSON · §1.3
The Language of Relations and Functions
Distinguish a relation from a function.
Before you begin
The Language of Sets
A set specifies which objects are included. Listing order and repetition do not change it.
{2,5}={5,2,2}5∈A asks about the number 5. {5}⊆A asks whether every element of {5} belongs to A.
5∈{2,5}; {5}⊆{2,5}Symbols
- R⊆A×B
- relation
- f:A→B
- function from A to B
Definitions and notation for this topic
Cartesian product
A × B
The set of all ordered pairs (a,b) with a ∈ A and b ∈ B.
The order of the entries in a pair matters.
Separate glossary example
{1} × {2,3} = {(1,2),(1,3)}
Function
f: A → B
Assigns exactly one element of B to each element of A.
Every allowed input must have exactly one output.
Separate glossary example
Let f:ℤ→ℤ be defined by f(x)=x+1. Then f(2)=3.
f(x)=x+1
Domain / codomain
f:A→B
In f:A→B, A is the domain of allowed inputs and B is the declared codomain. The image is the set of outputs actually attained.
Changing the codomain can change whether a function is onto. A/B is not notation for these two roles.
Separate glossary example
For f:ℤ→ℤ with f(x)=2x, both domain and codomain are ℤ, but f(ℤ) is the set of even integers.
f:ℤ→ℤ, f(x)=2x
Relation
R ⊆ A×B
A set of ordered pairs describing which objects are related.
A function is a special kind of relation.
The notation aRb means (a,b)∈R. A relation on A has R⊆A×A; reflexivity, symmetry and transitivity below concern that setting.
Separate glossary example
On ℤ, define aRb to mean a≤b.
Step by step
Step 1 / 6
A relation selects pairs
A relation from A to B is a set of selected pairs in A×B. An object may relate to several objects or none.
R⊆A×B
A relation does not itself require a unique output.
Worked example
A={1,2}, B={a,b}; R={(1,a),(2,a)}. Decide whether R is a function from A to B.
- Check the domain, not just the pairs: its inputs are 1 and 2.
- Input 1 occurs once, paired with a. Input 2 also occurs once, paired with a. Thus both existence and uniqueness hold.
- Both outputs belong to B. Therefore R is a function; two inputs sharing a and unused b are allowed.
Optional self-check
What does adding (1,b) violate?
Show answer
Input 1 has two different outputs, violating uniqueness.