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MathMod Prelim 3 - Coggle Diagram
MathMod Prelim 3
Functions and Relations
ordered pair
a set of inputs and outputs and represents a relationship between the two values
domain
refers to the set of possible input values (x)
range
the set of possible output values (y)
relation
a set of inputs and outputs
not all relations are functions.
Types
Empty Relations
When there’s no element of set X is related or mapped to any element of X, then the relation R in A is an empty relation, and also called the void relation. I.e R= ∅.
Universal Relations
R is a relation in a set, let’s say A is a universal relation because, in this full relation, every element of A is related to every element of A. i.e R = A × A.
Identity Relations
If every element of set A is related to itself only, it is called Identity relation. I={(A, A), ∈ a}
Inverse Relations
If R is a relation from set A to set B i.e R ∈ A X B. The relation R−1 = {(b,a):(a,b) ∈ R}.
Reflexive Relations
A relation is a reflexive relation If every element of set A maps to itself. I.e for every a ∈ A, (a, a) ∈ R.
Symmetric Relations
A symmetric relation is a relation R on a set A if (a, b) ∈ R then (b, a) ∈ R, for all a & b ∈ A.
Transitive Relations
If (a, b) ∈ R, (b, c) ∈ R, then (a, c) ∈ R, for all a,b,c ∈ A and this relation in set A is transitive.
Equivalence Relation
If a relation is reflexive, symmetric and transitive, then the relation is called an equivalence relation.
function
a relation with one output for each input.
All functions are relations
Types
One to one function or Injective function
A function f: P → Q is said to be one to one if for each element of P there is a distinct element of Q.
Many to one function
A function which maps two or more elements of P to the same element of set Q.
Onto Function or Surjective function
A function for which every element of set Q there is pre-image in set P
One-one correspondence or Bijective function
The function f matches with each element of P with a discrete element of Q and every element of Q has a pre-image in P.
Special functions
Constant Function
The c-value can be any number, so the graph of a constant function is a horizontal line
Identity Function
the x-value is the same as the y-value.
Linear Function
An equation written in the slope-intercept form is the equation and the graph of the function is a straight line.
Absolute Value Function
easy to recognize with its V-shaped graph. The graph is in two pieces and is one of the piecewise functions.
Inverse Functions
reverses the inputs with its outputs.