Function and Relation - Mathematics in the Modern World - GEED 10053 PDF

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Mathematics in the Modern World - GEED 10053...

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All functions are relations, but not all relations are functions. ▪

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A rule which uniquely associates elements of one set \((A)\) with the elements of another set \((B)\); each element in set \((A)\) maps to only one element in set \((B)\). Is a relationship where each element in the domain is related to one value only in the range by some rule. Is a set of ordered pairs (x,y) such that no two ordered pairs have the same x but different to y

Functions can be one-to-one relations or many-to-one relations. A many-to-one relation associates two or more values of the independent (input) variable with a single value of the dependent (output) variable. The domain is the set of values to which the rule is applied \((A)\) and the range is the set of values (also called the images or function values) determined by the rule. Example of a one-to-one function: \(y = x + 1\)

Example of a many-to-one function: \(y = x^{2}\)

Other examples: 1. f = {(1,3),(2,2),(3,4),(4,5)} 2. g = {(1,2),(2,5),(3,8)., (n,3n),…)} 3. h = {(2,5),(4,6),(5,11),(1,9),(0,8)}

Graphing Functions ▪

Using inputs and outputs listed in tables, maps, and lists, makes it is easy toplot points on a coordinate grid. Using a graph of the data points, you can determine if a relation is a function by using thevertical line test. If you can draw a vertical line through a graph and touch only one point, the relation is a function.

Take a look at the graph of this relation map. If you were to draw a vertical line through each of the points on the graph, each line would touch at only one point, so this relation is a function.

Special Functions ▪

Special functionsand their equations have recognizable characteristics.

Constant Function ▪ f(x)=cf(x)=c ▪ The c-value can be any number, so the graph of a constant function is a horizontal line. Here is the graph off(x)=4f(x)=4

Identity Function ▪ f(x)=xf(x)=x ▪

For theidentity function, the x-value is the same as the y-value. The graph is a diagonal line going through the origin.

Linear Function ▪ f(x)=mx+bf(x)=mx+b ▪ An equation written in theslope-intercept formis the equation of alinear function, and the graph of the function is a straight line. ▪ Here is the graph off(x)=3x+4f(x)=3x+4

Absolute Value Function ▪ f(x)=|x|f(x)=|x| ▪

Theabsolute value functionis easy to recognize with its V-shaped graph. The graph is in two pieces and is one of the piecewise functions.

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A rule which associates each element of set \((A)\) with at least one element in set \((B)\). A connection between the elements of two or more sets is Relation. The sets must be non-empty. A subset of the Cartesian product also forms a relationR.A relation may be represented either by Roster method or by Set-builder method.

EXAMPLE: Let A and B be two sets such that A = {2, 5, 7, 8, 10, 13} and B = {1, 2, 3, 4, 5}. Then, R= {(x, y): x = 4y – 3, x ∈A and y ∈B} (Set-builder form) R= {(5, 2), (10, 3), (13, 4)} (Roster form)

Types of Relations or Relationship Let us study about the various types of relations.

Empty Relation ▪

If no element of set X is related or mapped to any element of X, then the relation R in A is an empty relation, i.e, R = Φ. Think of an example of set A consisting of only 100 hens a poultry farm. Is there any possibility of finding a relation R of getting any elephant in the farm? No! R is a void or empty relation since there are only 100 hens and no elephant.

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Universal Relation ▪

A relation R in a set, say A is a universal relation if each element of A is related to every element of A, i.e., R = A × A. Also called Full relation. Suppose A is a set of all natural numbers and B is a set of all whole numbers. The relation between A and B is universal as every element of A is in set B. Empty relation and Universal relation are sometimes called trivial relation.

Identity Relation ▪

In Identity relation, every element of set A is related to itself only. I = {(a, a), ∈ A}. For example, If we throw two dice, we get 36 possible outcomes, (1, 1), (1, 2), … , (6, 6). If we define a relation as R: {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)}, it is an identity relation.

Inverse Relation ▪

Let R be a relation from set A to set B i.e., R ∈A × B. The relation R-1is said to be an Inverse relation if R-1from set B to A is denoted by R-1= {(b, a): (a, b) ∈ R}. Considering the case of throwing of two dice if R = {(1, 2), (2, 3)}, R-1= {(2, 1), (3, 2)}. Here, the domain of R is the range ofR-1and vice-versa.

Reflexive Relation ▪

If every element of set A maps to itself, the relation is Reflexive Relation. For every a ∈ A, (a, a) ∈ R.

Symmetric Relation ▪

A relation R on a set A is said to be symmetric if (a, b) ∈R then (b, a) ∈ R, for all a & b ∈ A.

Transitive Relation ▪

A relation in a set A is transitive if, (a, b) ∈R, (b, c) ∈ R, then (a, c) ∈ R, for all a, b, c ∈ A

Equivalence Relation ▪

A relation is said to be equivalence if and only if it is Reflexive, Symmetric, and Transitive. For example, if we throw two dices A & B and note down all the possible outcome.

Define a relation R= {(a, b): a ∈ A, b ∈ B}, we find that {(1, 1), (2, 2), …, (6, 6) ∈R} (reflexive). If {(a, b) = (1, 2) ∈R} then, {(b, a) = (2, 1) ∈ R} (symmetry). ). If {(a, b) = (1, 2) ∈R} and {(b, c) = (2, 3) ∈R} then {(a, c) = (1, 3) ∈R} (transitive)

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https://us.sofatutor.com/mathematics/algebra-1/functions-and-relations https://www.siyavula.com/read/maths/grade-12/functions/02-functions-02 https://www.toppr.com/guides/maths/relations-and-functions/types-of-relations/...