Relations and Functions

Introduction

The exploration of mathematics often intertwines concepts of beauty and aesthetics, termed by G. H. Hardy, who expresses that while defining beauty in mathematics may be complex, it is recognizable upon observation. The initial exploration of relations and functions, including domains, co-domains, and ranges, along with specific real-valued functions and their graphs, lays the foundation for understanding subsequent mathematical concepts. Relations in mathematics derive their meaning from everyday language, where two entities are said to be related if there exists a tangible connection between them.

Consider a scenario where set A represents all students in Class XII and set B consists of students in Class XI within the same institution. Various examples of relations can be drawn between these two sets, such as:
(i) ${(a, b) \in A \times B : a \text{ is a brother of } b}$
(ii) ${(a, b) \in A \times B : a \text{ is a sister of } b}$
(iii) ${(a, b) \in A \times B : a \text{ is older than } b}$
(iv) ${(a, b) \in A \times B : a \text{ has lower marks than } b}$
(v) ${(a, b) \in A \times B : a \text{ lives near } b}$.

Formally, a relation R from A to B is defined as an arbitrary subset of $A \times B$. If $(a, b) \in R$, we express this as a R b, irrespective of the explicit connection or definition that may link a and b. Functions, categorized as specialized relations, will be discussed extensively in this chapter, along with various types of relations, the composition of functions, invertible functions, and binary operations.

Types of Relations

Relations are fundamentally subsets of $A \times A$, with extremes being the empty set denoted by $\varnothing$ and the universal relation denoted by $A \times A$. For example, consider the relation R defined in the set $A = {1, 2, 3, 4}$ by defining a relation where $R = {(a, b) : a - b = 10}$. Since no pairs (a, b) satisfy this condition, $R$ is empty. In contrast, a relation $R'$ can be defined as $R' = {(a, b) : |a - b| \geq 0}$, which includes all element pairs, making it a universal relation.

Definition of Relations
  • Empty Relation: A relation $R$ in set A is considered an empty relation if

    R=∅⊆A×AR = \varnothing \subseteq A \times A

  • Universal Relation: A relation $R$ in set A is called a universal relation if every element in A is related to every element in A.

    R=A×AR = A \times A

Both the empty relation and the universal relation are informally referred to as trivial relations. For instance, let set A comprise all students in a boys' school. Showing that the relation $R = {(a, b) : a \text{ is a sister of } b}$, is indeed empty, confirms that R encompasses no valid pairs, thus demonstrating the characteristics of an empty relation. Conversely, the relation $R' = {(a, b) : |a - b| < 3}$ is established as a universal relation since the height difference among students is guaranteed to remain below three meters.

Representing Relations

When representing relations, familiar methods include:

  • Raster Method
  • Set Builder Notation
    For example, a relation $R$ defined as $R = {(a, b) : b = a + 1}$, may also be expressed in a conditional form, $a R b \iff b = a + 1$. As with earlier definitions, the expression $(a, b) \in R$ implies a relationship denoted by $a R b$.
Equivalence Relations

One pivotal relation in mathematics is the equivalence relation, which is conceptualized through three essential properties: reflexive, symmetric, and transitive.

  • Reflexive Relation: A relation $R$ in set A is reflexive if

    (a,a)∈R, for every a∈A.(a, a) \in R, \text{ for every } a \in A.

  • Symmetric Relation: A relation $R$ in set A is symmetric if

    (a1,a2)∈R  ⟹  (a2,a1)∈R, for all a1,a2∈A.(a_1, a_2) \in R \implies (a_2, a_1) \in R, \text{ for all } a_1, a_2 \in A.

  • Transitive Relation: A relation $R$ in set A is transitive if

    (a1,a2)∈R and (a2,a3)∈R  ⟹  (a1,a3)∈R, for all a1,a2,a3∈A.(a_1, a_2) \in R \text{ and } (a_2, a_3) \in R \implies (a_1, a_3) \in R, \text{ for all } a_1, a_2, a_3 \in A.

An equivalence relation $R$ can be defined as one that satisfies these three properties: reflexive, symmetric, and transitive.

Example of Equivalence Relation:
Let T be the set of all triangles, with relation R defined as

    R=(T1,T2):T1 is congruent to T2.R = {(T_1, T_2) : T_1 \text{ is congruent to } T_2}.
Proving R to be an equivalence relation involves demonstrating that it meets the aforementioned three criteria.

  • Reflexive: Every triangle is congruent to itself, $(T_1, T_1) \in R.$
  • Symmetric: If $(T_1, T_2) \in R$ implies $(T_2, T_1) \in R.$
  • Transitive: $(T_1, T_2) \in R$ and $(T_2, T_3) \in R$ implies $(T_1, T_3) \in R.$

Thus, R constitutes an equivalence relation.

Example 2: Let L be a collection of lines and relate R defined as

    R=(L1,L2):L1 is perpendicular to L2.R = {(L_1, L_2) : L_1 \text{ is perpendicular to } L_2}.
R here proves symmetric but fails reflexive and transitive qualities. If $(L_1, L_2) \in R$ implies $(L_2, L_1) \in R$, yet no line can be perpendicular to itself, affirming non-reflexive. Furthermore, if $L_1$ is perpendicular to $L_2$ and $L_2$ to $L_3$, then $L_1$ is parallel to $L_3$, thus not transitive.

Further Examples of Relations

Example 4: Show that the relation R in set ${1, 2, 3}$ defined by

    R=(1,1),(2,2),(3,3),(1,2),(2,3)R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3)}
is reflexive but neither symmetric nor transitive.
Solution: Reflexive as previous pairs confirm all diagonal entries are present. Non-symmetric since despite including $(1, 2)$, it lacks $(2, 1)$. Non-transitive as $(1, 2)$ and $(2, 3)$ do not result in $(1, 3)$.

Example 5: Consider the relation in set of integers:

R=(a,b):2 divides a−bR = {(a, b) : 2 \text{ divides } a - b}
This forms an equivalence relation, being reflexive, symmetric, and transitive. All even integers relate to zero, while odd integers relate to one, creating two distinct equivalence classes.

Types of Functions

Building upon prior learning with functions, including identity, constant, polynomial, and rational functions, the study extends to various types of functions defined by their characteristics: injectivity and surjectivity.

Definition of Functions
  • Injective Function (One-One): A function is defined as injective if distinct elements of its domain map to distinct elements of the codomain, i.e., for all $x_1, x_2 \in X$, if $f(x_1) = f(x_2)$, then $x_1 = x_2.$
    Function diagrams illustrate distinct mappings; functions that are not injective can have multiple points mapping to the same output.
  • Surjective Function (Onto): A function is surjective if every element of the codomain can be expressed as the image of some element of the domain. In mathematical terms, for every $y \in Y$, there exists an $x$ in $X$ such that $f(x) = y.$
Remark on Onto Functions

A function is onto if and only if the range of $f$ equals the entirety of the codomain Y.

  • Bijective Function: A function is bijective when it is both injective and surjective. Each element in the domain correlates uniquely to an element in the codomain without gaps or overlaps.
Example of Functions

Example 7: Let A represent a group of 50 students, with the function defined by their roll numbers:

f:A→N,f(x)=roll number of student xf: A \to N, \quad f(x) = \text{roll number of student } x

This function showcases injectivity since no two distinct students share the same roll number. However, it is not onto if student roll numbers are only defined from 1 to 50.

Example 8: The function $f: N \to N$, given as

f(x)=2x,f(x) = 2x,
This function emphasizes that while it remains injective, it is not surjective due to odd numbers in the natural set having no corresponding pre-image under f.

Example 9: Show that the function $f : R \to R$ given as:$f(x) = 2x$ is one-one (injective) and onto (surjective).
Solution: For injectivity:
Assume that $f(x_1) = f(x_2)$, which implies:
2x1=2x2  ⟹  x1=x2.2x_1 = 2x_2 \implies x_1 = x_2.
For surjectivity: Assuming y is any arbitrary real number, set, $y = f(\frac{y}{2}) = 2(\frac{y}{2})$. Therefore f is both injective and surjective.

Homework Exercises

  • Determine the characteristics of specified relations for reflexivity, symmetry, and transitivity.
  • Show relation characteristics under various set conditions to explore the equivalence and functions mappings.