“Relations and Functions” are the most important topics in algebra. Relations and functions – these are the two different words having different meanings mathematically. You might get confused about their difference. Before we go deeper, let’s understand the difference between both with a simple example. Show
An ordered pair is represented as (INPUT, OUTPUT): The relation shows the relationship between INPUT and OUTPUT. Whereas, a function is a relation which derives one OUTPUT for each given INPUT. Note: All functions are relations, but not all relations are functions.  In this section, you will find the basics of the topic – definition of functions and relations, special functions, different types of relations and some of the solved examples. What is a Function?A function is a relation which describes that there should be only one output for each input (or) we can say that a special kind of relation (a set of ordered pairs), which follows a rule i.e., every X-value should be associated with only one y-value is called a function. For example:
Let us also look at the definition of Domain and Range of a function.
Example: In the relation, {(-2, 3), {4, 5), (6, -5), (-2, 3)}, The domain is {-2, 4, 6} and range is {-5, 3, 5}. Note: Don’t consider duplicates while writing the domain and range and also write it in increasing order. Types of FunctionsIn terms of relations, we can define the types of functions as:
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Special Functions in AlgebraSome of the important functions are as follow:
What is the Relation?It is a subset of the Cartesian product. Or simply, a bunch of points (ordered pairs). In other words, the relation between the two sets is defined as the collection of the ordered pair, in which the ordered pair is formed by the object from each set. Example: {(-2, 1), (4, 3), (7, -3)}, usually written in set notation form with curly brackets. Relation RepresentationThere are other ways too to write the relation, apart from set notation such as through tables, plotting it on XY- axis or through mapping diagram. Video LessonRelations and Functions Class 12 MathsDifferent types of relations are as follows:
Let us discuss all the types one by one. Empty RelationWhen 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= ∅. For example, if there are 100 mangoes in the fruit basket. There’s no possibility of finding a relation R of getting any apple in the basket. So, R is Void as it has 100 mangoes and no apples. Universal relationR 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. It’s a full relation as every element of Set A is in Set B. Identity RelationIf every element of set A is related to itself only, it is called Identity relation. I={(A, A), ∈ a}. For Example, When we throw a dice, the total number of possible outcomes is 36. I.e (1, 1) (1, 2), (1, 3)…..(6, 6). From these, if we consider the relation (1, 1), (2, 2), (3, 3) (4, 4) (5, 5) (6, 6), it is an identity relation. Inverse RelationIf 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}. For example, If you throw two dice if R = {(1, 2) (2, 3)}, R-1 = {(2, 1) (3, 2)}. Here the domain is the range R-1 and vice versa. Reflexive RelationA relation is a reflexive relation iIf every element of set A maps to itself, i.e for every a ∈ A, (a, a) ∈ R. Symmetric RelationA symmetric relation is a relation R on a set A if (a, b) ∈ R then (b, a) ∈ R, for all a & b ∈ A. Transitive RelationIf (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 RelationIf a relation is reflexive, symmetric and transitive, then the relation is called an equivalence relation. Video LessonTypes of RelationsHow to Convert a Relation into a Function?A special kind of relation (a set of ordered pairs) which follows a rule i.e every X-value should be associated with only one y-value, then the relation is called a function. ExamplesExample 1: Is A = {(1, 5), (1, 5), (3, -8), (3, -8), (3, -8)} a function? Solution: If there are any duplicates or repetitions in the X-value, the relation is not a function. But there’s a twist here. Look at the following example: Though X-values are getting repeated here, still it is a function because they are associating with the same values of Y. The point (1, 5) is repeated here twice and (3, -8) is written thrice. We can rewrite it by writing a single copy of the repeated ordered pairs. So, “A” is a function. Example 2: Give an example of an Equivalence relation. Solution: If we note down all the outcomes of throwing two dice, it would include reflexive, symmetry and transitive relations. Then, throwing two dice is an example of an equivalence relation. Example 3: All functions are relations, but not all relations are functions. Justify. Solution: Let’s suppose, we have two relations given in below table Video LessonNumber of FunctionsLearn more on various Mathematical concepts with BYJU’S and enjoy practising through the most engaging videos. Download Free PDFs of Daily Practice Problems and Worksheet for Relations and Functions Concept Recommended VideosRelations and Types of RelationsFunctions – Introduction, Co-domain and RangeFrequently Asked Questions on Relations and FunctionsA relation from a set P to another set Q defines a function if each element of the set P is related to exactly one element of the set Q.
In math, a relation defines the relationship between sets of values of ordered pairs. The set of elements in the first set are called domain which is related to the set of the element in another set, which is called range. When
each input value of a function generates one and only output, it is called a function. Here, the input values are known as domain and output values are known as the range. If y = x + 2, is a function, then we have to put different values of x to
generate y values. Here, x is the input value and y is the output value. Suppose, if x = 0, then y =2, if x = 1, then y = 3, if x = -1, then y = 1, and so on. Now plot these values in a graph and join the points. What is it called when one input has one and only one output?When each input value has one and only one output value, that relation is a function. Functions can be written as ordered pairs, tables, or graphs. The set of input values is called the domain, and the set of output values is called the range.
When each input is assigned to exactly one output we say the relation is?A function is a relation where each input has exactly one output. In other words, a function is a specific kind of relation where none of the first entries are repeated. If we want to think about this in terms of domain and range, each number in the domain is related to exactly one number in the range.
Is a relationship that pairs each input value with exactly one output value?A function is a relation in which each possible input value leads to exactly one output value. We say “the output is a function of the input.” The input values make up the domain, and the output values make up the range.
What is a relation in which every member of the domain is paired with exactly one member of the range?A function is a special type of relation in which each element of the domain is paired with exactly one element of the range.
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A relation where every input in matches with exactly one output is called what
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