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 1 “Isa brother of" 2 "Isasister of"

• "Isa husband of

4 "sequal to"

• "Is greater than"

6 “Is tess than” Normally relations between twwo sets are indicated by an arrow coming from ore element of the first set going to the element of the other set

Relations Between Two Sets

Find relations between two sets The relation can be denoted as R= {(a, b): ais an clement ofthe first set, b is an clement of the second set} ‘Consider the following table pp pf]

PEPE PE

This the relation which can be writen asa set of ondared pts 1-3-6), (05,1). (0,212.4,

defining the set of all ordered pairs (x, y) such that y ~2x can be written symbolically as:

R= (Ky Y= 2X}

Relations Between Members in a Set Find velohons betvcen means ix eset Which ofthe following ordered pairs belong to the relation (3, yx}? (1,242. 1,(3.4),63,-).2,2, 68, 0,683 sohton (1,3,63,4)68,0),8-3)

Relations Pictorially

Damnsiaseretatons ihorealy For example the relation " is greater than ” involving numbers 1,2.3.4,8 and 6 where 1,3 and 5 belong to st A and 2,4 and 6 belong to st B canbe indoate as follows / \ / \

• \ 2 \

/ \ ae f 5 +——— —{ on Laeiio \ } \ aF \ a / Figuret. 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 That is R= { (@,b): acA, beB } anda is related to b) ‘The symbol e means belongs to or is a member of 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: Using a pictonal representation, — a —— —__—__/—js / < Se eo \ {3s Ss \ 6 }) ~7 ™\ ’ 2S / \ a oe ee, Se Aiso as a set of ordered pairs as ne 2,6), (3, 8), 6, 6),(2,10) and 6.0]

Example 1

• Draw an arrow diagram to illustrate the relation which conneets each clement of set A with its

square J

Let A= -2-1,0,1,2

ey }

Solution

x \ / % \ / Nea \ 2S ‘& ea / Peres Using the information given in example 1, write down the relation in set notation of ordered pairs. List the elements of ordered pairs.

Solution

bra

Reames Aswe, Let Rafa b):aeA, beB and } p-20

Use a pictorial diagram to ilustrate R

Solution

as a / \ / \ ,_ — Lo. \ aH ——-. \ F \, J

Example 4

Draw an arrow diagram to illustrate the relation " sTess than” IN. x /2: a \Y / ~~ / \ roo i a } \ p28 \,/ —s a, Na Draw a pictorial diagram between P and itself to show the relation relation " is a multiple of”

é

a3 = (KY): XEX, ¥EY

} i andyxe4-0 {fat which x is an integer less than 10 but greater than 2, then the following ordered pair does not belong to R (@) (3,-1) (b) (10,6), (€) (5,1), (d) (4,0) ( )

2 Let A}3.46.9)

Ifwe draw an arrow diagram between A and itself to show the relation js a multiple of How many arrows are counted? (@) 4 arrows (0) 8 arrows

• 6 arrows (a) 12 arrows ()

‘Consider the following pictorial diagram representing a relation R. ho R ~ ’ 1 _ ts k\ { 2+ ., ——_ +6 co} \ 2 } \ [ i" j \ J + 8/ From the figure above, the value of k is 4 (4 (12 @)3 ( ) Let the relation R be defined as

A R y~

fax f / +> Ff *« / { 2—T +2 ap -+. \ 9p \ 12 } \ J > a/

From the figure above, the value of kis @)4 (4 (12 3 ( )

factor of" How many elements from set a.are connected to 12 which is an element of set @1, 2 (3 WA «) Graph of a Relation A Graph of a Relation Represented by a Linear Inequality Draw a graph ofa relation represented by a linear inequality Given 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 relation Consider the following relation

mo OO [t+ 3 \ [3 __ +4 \ gf ON \ ‘ VA \ 6 Z£ ‘The above relation can be writfen aso set ordered Pairs

So 1 is related to 3. 2 is related to 3 and so on , there fore (4.3), (2.3), (24), (4,5) a all on the graph ‘The graph of R is shown the following diagram( x-y plane}, Y al 5 x 4 x xx ™ po x a

Example S

Solved:

dcaw a graph to ilustrate the relation “is a factor of”

Solution

The relation “is a factor of “can be written as a set of ordered pairs as R-{22) (2.4), 26) 8.3) (3.6), (4.4), 65)} Note that some relations have graphs representing special figures like straight lines or curves,

Example 6

Solution

draw its araph we must have at leat two points through whieh the line passes.

Graph; rth J were a / aot x 4a a2Advr23a456 joa éo4

Example7

Solution

Greph i t ia & aa i tt ttt x S43 24/123 45 NB: When the relation is given by an equation such as y~ F(x), the domain isthe set containing x valuas satisfying the equation and the range is the set of y-values satisfying the given equation

Test Yourself

• The relation R is given by y = 2x + 3, write Ras the set of ordered pairs where x is an

integer such that -1<x< 11

• Leta 5.18, 20} and B ={ 4, 10, 17, 19}
• Draw an arrow diagram to show the relation ‘is less than “
• Draw the graph of this relation.

Quiz.

• The relation whose graph is a straight line passing through (1,1) and (2,3) is

• One of tho paints through which the graph of the relation x_— y =5 passes is

(2)@.5) ©) 23) (©) (3.2) @)@,-5) ()

• Gen that Ac 32×25 } and Bele £9). ‘an element from set A is

mapped onto an element in set 8 by the relation “is less than’ iA and B are sets of integers, what is the greatest integer in set A can be mapped onto an element from sot 8 7 (a3 @)4 ()5 G4 ( )

@) [x:x>t} (by{ x2xis any real number } = [x:xzo} { sx<j] e Y

• What is the range of the relation R = { txyly=x }

avo} m{ rye (o{y:¥ 8 any tea! number} (8) {v3720 () Domain and Range of a Relation ‘The Domain of Relation State she domain of relanion Domain: The domain ofa function is the set ofall possible input values (ofien the "x" variable), whieh produce a valid output from a particular funetion, I is the set ofall real numbers for which a Function is mathematically defined ‘The Range of'a Relation State she range of a relation

Range: The range is the sct of all possible output values (usually the variable y, or sometimes expressed as ffx), which result from using a particular function IPR 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 isthe Co-domain or Range Note that each member of set A must be mapped to at least one element of set Band each member of sot B must be an image of a least one element in set A.

Consider the following relation aN A \ “a an 2 f poe \ { 2 ae { \ 4 / \ cf \ / Kf For the above relation, the domain is {1,3} while the range is{ a.b}

Example 8

Find the domain and range of the relation R2” is less than’ Pa “RO > f toh fo\ / pe SS \ to \ 10 — a a / \ / \ J From the pictorial representation of the relation R above, the Domain is {1.3.4} and the Range Is (4,8)

Example 9

Aswe, and -2<x<8 J Where is the relation and both x and y are integers. State the domain and range of R

Solution

Domain +{ x: -2<x<8 }

• Let A= {35,79 } and B= {1,468} , find the domain and range of the relation “is greater

than on sets A and B

• Let Z={ Triangle, quadrateral, pentagon, hexagon}

and W 123,45}. Find the domain and range ofthe relation between Z and W that connects each polygon with the number of is sides.

• State the domain and range of the following relation

gies R s~ / \ / \ 2 } we cael }

\ F Le

Draw the pictorial diagram to illustrate the relation “is less than or equal to* and state its domain and range Inequalities: The equations involving the signs <, <,> or are called inequalities

24 x is greater than 3 XE 2 xis less or equal to 2 2x is greater or equal to? x>y x is greater or than y ete Inequalities can be shown on a number ine as m the following

Examples;

1 x33 10123 4 5 6 2 -2sxs5 -——_______.

Sa SS Se eS

3240123456789 10 Inequalities involving two variables: It the inequality involves two variables itis 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 sand > The line drawn separates the x-y plane into two patts/regions The region satisfying the given inequality is shaded and before shading it must be tested by

Example 10

• Draw the graph of the relation R= {(x, y): xy}

Graph z KK

• Draw the graph of the relation R= {(x,y ): x + y> 0}

a

• White down the inequality for the relation given by the following graph

iv 1 kK \

ESI LOLS * &

LN \h ale 4, Draw a graph of the inequality for the relation x >-2 and shade the required region Domain and Range from the graph

Definition: Domain is the sot ofall x values that satisfy the given equation oF inequality

Similarly Range is the set of ally value satisfying the given equation or inequality

Example 11

• Consider the following graph and state its domain and range

ay rel CI i Ay a es ce ee 543244 123 4 Solation

Example 12

a ay 432 4 412345 al

Solution;

Inverse of a Relation The Inverse of a Relation Pictorially Explain she Inverse of a relation pictorial Ir there is relation between two sets A and B interchanging A and B gives the inverse of the relation.

IER is the retation, then its inverse is denoted by R"*

• If the relation is shown by an arrow diagram then reversing the direction of the arrow

ives its inverse

• If the relation is given by ordered pair ( x, y).. then inter changing the variables gives

Example 13

aN —~ 8 / 2 ———— oN / 3. 6 OS XS ae The inyerse ofthis relation is“ is « multiple of (SS { ox 3 \ \ 10—<{— 5} foo / Inverse of a Relation Find inverse ofa relation

Example 14

B

Solution

R'is obtained by inter changing the variables x and y. X+32y Y+32x yex3 oR Guy): wey

Or R*={(xy) xasy}

Example 18

Find the inverse ofthe relation

Solution

After interchanging the variable x and y, the equation

soRT=(xy)ey= lex

element from set A on to the element in set B

3 Find the inverse of the following relation represented in pictorial diagram Fy / nae \ —S— \ ae ae oe i a \ po A Graph of the Inverse of a Relation Draw a graph of the inverse of a relation Lf ANY horizontal line intersects your orginal function in ONLY ONE location, your function

ea ieerdp! 0) dueeeees i fi el fl