1.2 Relation  

Definition

"A relation from a set A to a set B is a set of ordered pairs (a, b) where a ∈ A and b ∈ B. If A = B, then the relation is said to be on A."

"For example, ‘is greater than’ is a relation between numbers. ‘Is a parent of’ is a relation between people."

Properties of Relations

"Relations can have several important properties that help us classify and understand them."

Reflexive: A relation R on a set A is reflexive if every element in A is related to itself. In other words, for all a ∈ A, (a, a) ∈ R  

"Think of ‘is equal to’. Every number is equal to itself."

Symmetric: A relation R on a set A is symmetric if whenever (a, b) ∈ R, then (b, a) ∈ R.

"Think of ‘is a sibling of’. If person A is a sibling of person B, then person B is a sibling of person A."

Transitive: A relation R on a set A is transitive if whenever (a, b) ∈ R and (b, c) ∈ R, then (a, c) ∈ R  

"Think of ‘is an ancestor of’. If person A is an ancestor of person B, and person B is an ancestor of person C, then person A is an ancestor of person C."

Irreflexive: A relation R on a set A is irreflexive if no element in A is related to itself. In other words, for all a ∈ A, (a, a) ∉ R  

Closures of Relations

"Sometimes a relation doesn’t have a certain property, but we can add ordered pairs to it to make it have that property. This is called taking the closure of the relation."

Reflexive Closure: The reflexive closure of a relation R is the smallest reflexive relation that contains R.

Symmetric Closure: The symmetric closure of a relation R is the smallest symmetric relation that contains R.

Transitive Closure: The transitive closure of a relation R is the smallest transitive relation that contains R.

Equivalence Relations and Partitions

"An equivalence relation is a relation that is reflexive, symmetric, and transitive. Equivalence relations are very important because they partition a set into disjoint subsets called equivalence classes."

"An equivalence class is a set of all elements that are related to each other under the equivalence relation."

"A partition of a set is a collection of disjoint subsets whose union is the entire set."

"Therefore, an equivalence relation on a set creates a partition of that set, where each part of the partition is an equivalence class."

Classroom Activities

Relation Property Identification:

"I’ll give you some relations, and you have to identify which properties they have (reflexive, symmetric, transitive)."  

Equivalence Relation Construction:

"I’ll give you a set, and you have to construct an equivalence relation on that set, and then identify the equivalence classes."

 1.3 Functions:  

Definitions

"Formally, a function from a set A to a set B is a relation that assigns to each element x in A exactly one element y in B. We write this as f: A → B, where A is the domain and B is the codomain."

"In simpler terms: For every input, there is only one output. No input can have multiple outputs.

"Let’s say our machine is a ‘double it’ machine. If you put in 2, it outputs 4. If you put in 5, it outputs 10. Each input has only one output. That’s a function!"

Types of Functions

"Now, let’s explore some special types of functions."

One-to-One: A function f: A → B is one-to-one if different elements in A map to different elements in B. In other words, if x₁ ≠ x₂, then f(x₁) ≠ f(x₂).

"Our ‘double it’ machine is one-to-one. Every input gives a unique output."

"Example: f(x) = x + 5 is one-to-one. If you get a particular output, you know exactly what the input was."

Onto: A function f: A → B is onto if every element in B is the image of some element in A. In other words, for every y in B, there exists an x in A such that f(x) = y.

"Imagine a machine that always outputs even numbers. If our codomain is all even numbers, then it’s onto, because every even number can be an output. But if our codomain is all integers, it’s not onto, because we’ll never get an odd number as an output."

"Example: If f(x) = 2x and B is the set of all even numbers, then f is onto."

Bijection: A function f: A → B is a bijection if it is both one-to-one and onto.

"A bijection is a perfect pairing between the elements of two sets. Every input has a unique output, and every output has a unique input."

"Our ‘double it’ machine, where the codomain is only even numbers, is a bijection."

Function Composition

"Function composition is when you apply one function to the result of another function (Composition of Functions, 2018). It’s like having two machines in a row. You put something into the first machine, it does its thing, and then the output goes into the second machine."

"If we have two functions, f: A → B and g: B → C, then the composition of f and g is denoted as g(f(x)) or (g ∘ f)(x)."

·         "Let’s say f(x) = x + 1 and g(x) = x². Then g(f(x)) = g(x + 1) = (x + 1)²"