Normally relation deals with matching of elements from the first set called
DOMAIN with the element of the second set called RANGE. Relations A relation “R” is the rule that connects or links the elements of one set with the elements of the other set.Some examples of relations are listed below:
- “Is a brother of “
- “Is a sister of “
- “Is a husband of “
- “Is equal to “
- “Is greater than “
- “Is less than “
Normally
relations between two sets are indicated by an arrow coming from one
element of the first set going to the element of the other set.Relations Between Two SetsFind relations between two setsThe relation can be denoted as:R = {(a, b): a is an element of the first set, b is an element of the second set}Consider the following table
This
is the relation which can be written as a set of ordered pairs {(-3,
-6), (0.5, 1), (1, 2), (2, 4), (5, 10), (6, 12)}. The table shows that
the relation satisfies the equation y=2x. The relation R defining the
set of all ordered pairs (x, y) such that y = 2x can be written
symbolically as:R = {(x, y): y = 2X}.Relations Between Members in a SetFind relations between members in a setWhich of the following ordered pairs belong to the relation {(x, y): y>x}?(1, 2), (2, 1), (-3, 4), (-3, -5), (2, 2), (-8, 0), (-8, -3).Solution.(1, 2), (-3, 4), (-8, 0), (8,-3).Relations PictoriallyDemonstrate relations pictoriallyFor
example the relation ” is greater than ” involving numbers 1,2,3,4,5
and 6 where 1,3 and 5 belong to set A and 2,4 and 6 belong to set B can
be indicate as follows:-
This kind of relation representation is referred to as pictorial representation.Relations
can also be defined in terms of ordered pairs (a,b) for which a is
related to b and a is an element of set A while b is an element of set
B.
For
example the relation ” is a factor of ” for numbers 2,3,5,6,7 and 10
where 2,3,5 and 6 belong to set A and 6,7 and 10 belong to set B can be
illustrated as follows:-
Example 1<!–
[if !supportLists]–>1. Draw an arrow diagram to illustrate the
relation which connects each element of set A with its square.
Solution
Example 2Using
the information given in example 1, write down the relation in set
notation of ordered pairs. List the elements of ordered pairs.
Example 3As we,
Solution;
Example 4Let X= {2, 3, 4 } and Y= {3 ,4, 5}Draw an arrow diagram to illustrate the relation ” is less than”
Exercise 1Let P= {Tanzania, China, Burundi, Nigeria}Draw a pictorial diagram between P and itself to show the relation“Has a larger population than”2. Let A = 9,10,14,12 and B = 2,5,7,9 Draw an arrow diagram between A and B to illustrate the relation ” is a multiple of”3.Let A = mass, Length, time andB = {Centimeters, Seconds, Hours, Kilograms, Tones}Use the set notation of ordered pairs to illustrate the relation “Can be measured in”4.
A group people contain the following; Paul Koko, Alice Juma, Paul
Hassan and Musa Koko. Let F be the set of all first names, and S the set
of all second names.Draw an arrow diagram to show the connection between F and S5. Let R={ (x, y): y=x+2}Where x∈A and A ={ -1,0,1,2}and y∈B, List all members of set BExercise 21. Let the relation be defined
Consider the following pictorial diagram representing a relation R.
Let the relation R be defined as
A relation R on sets a and B where A = 1,2,3,4,5 and B = 7,8,9,10,11,12 is defined as ” is a factor of “
Graph of a RelationA Graph of a Relation Represented by a Linear InequalityDraw a graph of a relation represented by a linear inequalityGiven
a relation between two sets of numbers, a graph of the relation is
obtained by plotting all the ordered pairs of numbers which occur in the
relationConsider the following relation
The graph of R is shown the following diagram( x-y plane).
Example 5Solved:
Note that some relations have graphs representing special figures like straight lines or curves.Example 6Draw the graph for the relation R= {(x, y): y = 2x +1} Where both x and y are real numbers.SolutionThe
equation y = 2x +1 represents a straight line, this line passes throng
uncountable points. To draw its graph we must have at least two points
through which the line passes.
Graph;
Example 7Let A = {-2,-1,0, 1, 2 } and B ={0,1,2,3,4}Let the relation R be y= x2, where x ∈A and y∈B. Draw the graph of RSolution
NB:
When the relation is given by an equation such as y = f (x), the domain
is the set containing x- values satisfying the equation and the range
is the set of y-values satisfying the given equation.Exercise 3Test Yourself:
Quiz.
Domain and Range of a RelationThe Domain of RelationState the domain of relationDomain:
The domain of a function is the set of all possible input values (often
the “x” variable), which produce a valid output from a particular
function. It is the set of all real numbers for which a function is
mathematically defined.The Range of a RelationState the range of a relationRange:
The range is the set of all possible output values (usually the
variable y, or sometimes expressed as f(x)), which result from using a
particular function.If
R is the relation on two sets A and B such that set A is an independent
set while B is the dependent set, then set A is the Domain while B is
the Co-domain or Range.Note
that each member of set A must be mapped to at least one element of set
B and each member of set B must be an image of at least one element in
set A.Consider the following relation
Example 8Let P = 1,3,4,10 and Q = 0,4,8Find the domain and range of the relation R:” is less than”
Example 9As we,
Exercise 41. Let A = { 3,5,7,9 } and B = {1,4,6,8 } , find the domain and range of the relation “is greater than on sets A and B
4. Let X ={3, 4, 5, 6} andY ={2, 4, 6, 8}Draw the pictorial diagram to illustrate the relation “is less than or equal to‘ and state its domain and rangeInequalities:The equations involving the signs < , ≤, > or ³ are called inequalitiesEg. x<3 x is less than 3x>3 x is greater than 3x≤ 2 x is less or equal to 2x³ 2 x is greater or equal to 2x > y x is greater or than y etcInequalities can be shown on a number line as in the following
Inequalities involving two variables:If
the inequality involves two variables it is treated as an equation and
its graph is drawn in such a way that a dotted line is used for > and
< signs while normal lines are used for those involving ≤ and ≥.The line drawn separates the x-y plane into two parts/regionsThe
region satisfying the given inequality is shaded and before shading it
must be tested by choosing one point lying in any of the two regions,Example 101. Draw the graph of the relation R = {(x, y): x>y}Solution:x>y is the line x =y but a dotted line is used.Graph
If you draw a graph of the relation R = {(x,y ) : x < y} , the same line is draw but shading is done on the upper part of the line.Exercise 51. Draw the graph of the relation R = {(x,y ): x + y > 0}2 .Draw the graph of the relation R = {( x ,y ) : x – y ³ -2}3. Write down the inequality for the relation given by the following graph
4. Draw a graph of the inequality for the relation x >-2 and shade the required region.Domain and Range from the graphDefinition: Domain is the set of all x values that satisfy the given equation or inequality.Similarly Range is the set of all y value satisfying the given equation or inequalityExample 111. Consider the following graph and state its domain and range.
Solution
Example 12State the domain and range of the relation whose graph is given below.
Inverse of a RelationThe Inverse of a Relation PictoriallyExplain the Inverse of a relation pictoriallyIf there is a relation between two sets A and B interchanging A and B gives the inverse of the relation.If R is the relation, then its inverse is denoted by R-1
- If the relation is shown by an arrow diagram then reversing the direction of the arrow gives its inverse
- If the relation is given by ordered pair ( x, y) , then inter changing
the variables gives inverse of the relation, that is (y,x) is the
inverse of the relation. So domain of R = Range of R -1 and range of R =
domain of R-1
Example 131.
The inverse of this relation is “ is a multiple of “
Inverse of a RelationFind inverse of a relationExample 14Find the inverse of the relation R ={ ( x, y):x+ 3 ³ y}SolutionR-1 is obtained by inter changing the variables x and y.
Example 15Find the inverse of the relationR ={ ( x , y ): y = 2x }SolutionR ={( x , y ): y = 2x }After interchanging the variable x and y, the equationy = 2x becomes x = 2yor y = ½ xso R-1 = ( x, y ) : y = ½ xExercise 61
.Let A = 3,4,5 and B ‘= 1,4,7 find the inverse of the reaction “ is
less than “ which maps an element from set A on to the element in set B2 .Find the inverse of the relation R = {( x ,y ) : y > x – 1}3 .Find the inverse of the following relation represented in pictorial diagram
4 .State the domain and range for the relation given in question 3 above5. State the domain and range of the inverse of the relation given in question 1 above.A Graph of the Inverse of a RelationDraw a graph of the inverse of a relationUse thehorizontal line testto determine if a function has aninverse function.If
ANY horizontal line intersects your original function in ONLY ONE
location, your function has an inverse which is also a function.The functiony= 3x+ 2, shown at the right, HAS aninverse functionbecause it passes the horizontal line test.
