Ordered Pairs: Relations and Functions

Relations and Functions are important Mathematics concepts for every Class 10 student. Both the terms have different meanings and approaches. Wondering what exactly are there? We are here to guide you with the same.

Suppose there are two variables: A and B. Order is represented as (A, B), and relation representation is given in terms of how the two variables are related to each other. On the other hand, function derives the relation of the first variable to the second variable and vice versa.

Note: All functions are relations, but not all relations are functions.

Relations and Functions

A fixed order always exists in between two objects or elements. The equality of two ordered pairs is given as.
(?1,?1)=(?2,?2)⇔?1=?2a1,b1=a2,b2⇔a1=a2 and ?1=?2

A function is defined as a relationship describing only one type of output for each input. You can say that special kinds of relations exist between the ordered pairs which follow a rule, i.e., every x-value should be associated with only one y-value named as a function. Functions contain two aspects: domain and range.

The domain is the set of all first values within the paired set of order. The range is the set of second values which exists in the ordered pair.

Consider the relation- {(-2, 3), {4, 5), (6, -2), (4, 3)}.
The domain is {-2, 4, 6} and range is {-2, 3, 2}.

Special Functions in Algebra

Know some of the important functions when it comes to Algebra Class 10.

• Constant Function: ConstaConstant Functions have a continuous set of ranges or values. The value stays constant over time. Graphically, a constant function is a straight horizontal line.
  
• Identity Function: The identity function returns the same value as the original argument. For example, suppose if you consider f as a function, then the identity relation for argument x is represented as f(x) = x for all values of x.
  
• Linear Function: Linear function is the type of function that contains all the variables without any exponents.
  
• Absolute Value Function: A function representing the actual value of an integer via a number line is an absolute value function. The absolute value of a number is always positive.
  
• Inverse Functions: Function returning the original value for which a given output is there is an inverse function. Suppose, if f(x) is a function that gives output y, then the inverse function of y, i.e. f^-1(y), will return the value x.

Special Relations in Algebra

Here are some of the special types of relations in Algebra that students of Class 10 must know.

• Empty Relations: Also known as void relation, empty relation is a relation that contains no element. It is represented as R= ∅.
  
• Universal Relations: Consider A is a universal relation because every element is related to each other. i.e R = A × A.
  
• Identity Relations: If every element of A is present in its own element, it is named an identity relation. For example, when you throw a dice, there are 36 possible outcomes.  Out of these, (1, 1), (2, 2), (3, 3) (4, 4) (5, 5) (6, 6), are identity relation.
  
• Inverse Relations: If R is a relation from set A to set B i.e R ∈ A X B. The relation R= {(b,a):(a,b) ∈ R}. For Example, if you throw two dice if R = {(2, 3) (3, 4)}, then R’= {(3, 2) (4, 3)}. Here the domain range of R gets inversed and vice versa.

• Reflexive Relations: When every element of A maps to itself, it is named reflexive relations. For instance, a ∈ A, (a, a) ∈ R.
  
• Symmetric Relations: A symmetric relation is defined as 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 Relations: Equivalence Relations are reflexive, symmetric, and transitive.

Functions & Relations: Know Important Concepts

Let ?A and ?B be two non-empty sets. Then a relation ?f from ?A to ?B is a function, if

• For every, ?∈? there exists ?∈? such that (?,?)∈?
  
• (?,?)∈?a, b∈f and (?,?)∈?⇒?=?a,c∈f⇒b=c.
  
• In other words, f is a function from ? to ? if each element of ? appears in some ordered pair in ? and no two ordered pairs in ? have the same first element.
  
• If (?,?)∈?, then ? is called the image of ? under ?.
  
• A function ? from a set ? to a set ? is a rule associating elements of set ? to elements of set ? such that every element in set ? is associated with a unique element in set ?.
  
• Here the set ? is called the domain of ? and the set ? is called its co-domain.
  
• The range of a function ? is the set of images of elements in the domain.

Know all the important formulas with reference to the Algebra of Real Functions.

Questions on Functions and Relations

Now that you know all the important formulas and highlights, given below are some of the questions on Relations and Functions that you can all solve on your own.

Q.1: Is A = {(1, 5), (1, 5), (3, -8), (3, -8), (3, -8)} a function?

Q.2: Give an example of an Equivalence relation.

Q.3: How to graph a function?

Q.4: Let P = {(x, y) : x²+y²=1, x, y ∈ R}. Then, P is
(a) Reflexive
(b) Symmetric
(c) Transitive
(d) Anti-symmetric

Q.5: Let S be the set of all real numbers. Then, the relation R = {(a, b) : 1 +ab > 0} on S is
(a) Reflexive and symmetric but not transitive
(b) Reflexive and transitive but not symmetric
(c) Symmetric, transitive but not reflexive
(d) reflexive, transitive and symmetric

We hope that this comprehensive article on ‘Relations and Functions’ has been of assistance and provided you with insights into the algebra concepts of Relations and Functions.

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