Relations and Functions

Notes on Relations and Functions

1.1 Introduction

• Relational Concepts: The notion of relations and functions is essential in mathematics. A relation connects elements of two sets, defined mathematically as a subset of the Cartesian product of two sets. For example, if A and B are sets of students from two different classes, a relation from set A to set B includes pairs where a member from A is related to a member from B through specific criteria (sibling relations, age comparison, etc.).

• Function: A function is a special kind of relation where each element in the domain maps to exactly one element in the co-domain. This chapter builds on concepts learned in previous classes and delves deeper into definitions and characteristics of various types of functions.

1.2 Types of Relations

• Types of relations include:

  • Empty Relation (R): No elements relate to one another, denoted as R = φ.
  • Universal Relation (R'): Every element of set A relates to every element of A, represented as R = A × A.
  • Reflexive Relation: A relation is reflexive if every element relates to itself, meaning (a, a) ∈ R for every a in A.
  • Symmetric Relation: A relation R is symmetric if (a, b) ∈ R implies (b, a) ∈ R.
  • Transitive Relation: A relation R is transitive if (a, b) ∈ R and (b, c) ∈ R implies (a, c) ∈ R.
  • Equivalence Relation: A relation that is reflexive, symmetric, and transitive. Example: the relation of congruence among triangles.

• Equivalence Classes: These are subsets containing elements that are related under an equivalence relation.

1.3 Types of Functions

• Definitions and properties:
  • Injective Function (One-One): A function is injective if distinct elements in the domain map to distinct elements in the co-domain. Mathematically, f(x1) = f(x2) implies x1 = x2.
  • Surjective Function (Onto): A function is surjective if every element in the co-domain is covered by at least one element from the domain. Hence, the range equals the co-domain.
  • Bijective Function: A function is bijective if it is both injective and surjective, allowing for an exact one-to-one correspondence between domain and co-domain.

1.4 Composition of Functions and Invertible Functions

• Function Composition: The composition of two functions f and g, denoted as g ∘ f, is defined by (g ∘ f)(x) = g(f(x)). The resultant function encompasses the output of f as the input for g.
• Invertible Functions: A function f is invertible if an inverse function g exists such that g ∘ f = I (identity function on X) and f ∘ g = I on Y, indicating that applying both functions in succession returns the original element.

Key Theorems and Results

• Any one-one function from a finite set to itself is onto and vice versa. This property does not necessarily hold for infinite sets.
• Functions can be evaluated based on their actions (e.g., multiples, additions). Different cases, such as the Greatest Integer Function and Modulus Function, may demonstrate injectivity/surjectivity.
• The concepts of function notation and properties transitioned historically from early mathematicians to modern definitions emphasizing set theory.

1. Relation: A subset of a Cartesian product A × B showing connections between A and B.
2. Function: A special relation where each input has a unique output.
3. Empty Relation: The relation containing no elements (R = φ).
4. Universal Relation: Contains all possible pairs (R = A × A).
5. Equivalence Relation: A reflexive, symmetric, and transitive relation.
6. Injective Function: Maps distinct inputs to distinct outputs.
7. Surjective Function: Every element of the co-domain is covered by the function.
8. Bijective Function: Both injective and surjective (one-to-one correspondence).
9. Function Composition: Combining functions to form new mappings.
10. Invertible Function: A function that has a unique inverse running the operation in reverse.