relations and functions

Relation (informal definition)

a set A may be related to set B if A is a subset of B, not a subset of B, or if A and B have at least one element in common

Number Relation Example

a number x may be related to a number y if x < y, x is a factor of y, or x² + y² = 1

Definition of Relation (example)

x R y means x is related to y under a given condition

Example Relation Condition

x R y if and only if x < y for A = {1,2,3} and B = {2,3,4}

Related Ordered Pairs Example

{(1,2), (1,3), (1,4), (2,3), (2,4), (3,4)}

Non-related Example

1 R 1 and 3 R 2 are not relations under x < y

Cartesian Product Example

A × B = {(1,2), (1,3), (1,4), (2,2), (2,3), (2,4), (3,2), (3,3), (3,4)}

Relation Definition (Peirce)

a relation is the totality of ordered pairs whose elements are related by a given condition

Formal Definition of Relation

a relation from set X to Y is the set of ordered pairs (x, y) where each x in X corresponds to at least one y in Y

Formal Definition 2

a relation R from A to B is a subset of A × B

Domain of a Relation

the set A in a relation R from A to B

Co-domain of a Relation

the set B in a relation R from A to B

Relation Notation

x R y means (x, y) ∈ R; x R̸ y means (x, y) ∉ R

Domain Example

{0, 1, 2, 3, 4, 5} for relation {(0,-5),(1,-4),(2,-3),(3,-2),(4,-1),(5,0)}

Co-domain Example

{-5, -4, -3, -2, -1, 0}

Integer Relation Example

x R y if x/y is an integer for A = {1,2}, B = {1,2,3}

Cartesian Product (integer example)

A × B = {(1,1),(1,2),(1,3),(2,1),(2,2),(2,3)}

Integer Relation Subset

R = {(1,1),(1,3),(2,2)}

Arrow Diagram Definition

represent elements of A and B as points and draw arrows from x to y if x is related to y

Arrow Diagram Example

relation {(1,2),(0,1),(3,4),(2,1),(0,-2)}

Arrow Diagram Domain

{0, 1, 2, 3}

Arrow Diagram Co-domain

{-2, 1, 2, 4}

Relation S Example

S = {(1,3),(1,5),(2,3),(2,5),(3,5)} where x < y for sets A = {1,2,3}, B = {1,3,5}

Relation T Example

T = {(2,1),(2,5)}

Definition of Function

a relation in which every input is paired with exactly one output

Function Unique Output Rule

no two distinct ordered pairs have the same first component

Domain of a Function

the set of all x-values

Range or Co-domain of a Function

the set of resulting y-values

Formal Function Definition

a function F from A to B assigns each input exactly one output

Operations of Functions: Addition

(f + g)(x) = f(x) + g(x)

Addition Example

f(x) = 2x + 1 and g(x) = 3x + 2

Operations of Functions: Multiplication

(f × g)(x) = f(x) × g(x)

Multiplication Example

f(x) = 2x + 1 and g(x) = 3x + 2

Operations of Functions: Division

(f ÷ g)(x) = f(x) / g(x)

Division Example

f(x) = 2a + 6b and g(a,b) = a + 3b

Operations of Functions: Composition

(f ∘ g)(x) = f(g(x))

Composition Example

f(x) = 2x + 1 and g(x) = 3x + 2