Understanding Mathematical Relations through Cartesian Products and Ordered Pairs

DAV ZONE-1 ODISHA CHAPTER :RELATIONS (CLASS XI)

Previous Learning Student has to know Cartesian products of two nonempty sets that knowledge is greatly required to represent all types of relations, whether it could be a; Social Relations ; e,g: Ram is the father of Lava. Mathematical Relations ; e,g: y is double of x. Biological Relations ; e,g: interactions between two organisms such as predator-prey. Planetary Relations ; e,g: Relation between nine planets of our solar system. Physical Relations ; e,g: S= gt2

LEARNING OBJECTIVES 1.To represent above mentioned relations in mathematical form using set theory & cartesian product. 2. To determine it by arrow diagram. 3. To know its domain, co-domain and range.. 4. To know image and pre-image. 5. To know its representations. i. Roaster form :- Where element gets importance in associations. E,g:- Rama and Hari Clear your school dues. ii. Set builder Form :- Where relation gets importance. e,g:- Who have not cleared please clear the dues.

Ordered pair (language of every relation):- An ordered pair consists of two objects or elements in a given fixed order. For example : if A and B are any two sets, then by an ordered pair of elements we mean the pair (a,b) in that order, where a A and b B .

Equality of ordered Pairs : Two ordered pairs (a,b) and (c,d) are said to be equal if a = c and b=d . In general (a,b) (b,a). Example : Find the value of a and b, if (2a-3,b-1) = (a+2,9) Solution : By the definition of equality of ordered pairs we have (2a-3,b-1) = (a+2,9) 2a-3 = a + 2 and b-1=9 a = 5 and b=10.

CARTESIAN PRODUCT OF SETS : If A,B, are two non-empty sets, then their Cartesian product, denoted by A B is the set of all ordered pairs (a,b) such that a A and b B and is read as A cross B. Thus A B = {(a,b) : a A and b B } Example : If A = { 1,2,3} , B = {a,b} then A B = {(1,a),(1,b),(2,a),(2,b),(3,a),(3,b)} Example : If A = { 1,2} then A A A = {(1,1,1),(1,1,2),(1,2,1),(1,2,2),(2,1,1), (2,1,2),(2,2,1),(2,2,2)}

Note : If A and B are finite sets, then n(A B)=n(A).n(B) Example: If a set A contains 3 and set B contains 2 elements then the set A B contains 6 elements. Note : n{(A B) (C D) } = n{(A C) (B D)} Example: If two sets A & B are having 4 elements in common, then the no. of elements common to each of the sets A B & B A is 16 because n {(A B) (B A) } = n{(A B) (B A)} = n(A B) n(B A) = 4 4 = 16

Graphical Representation of cartesian product of sets : Let A and B any two non-empty sets. To represent A B graphically , draw two mutually perpendicular lines, one horizontal and other vertical. On the horizontal line, represent the elements of sets A and on the vertical line, the elements of set B.If a A and b B , draw a vertical line through a and a horizontal line through b . These two lines will meet in a point which will denote the order pair (a,b). In the way mark points corresponding to each ordered pair in A B. The set of points so obtained represents A B graphically.

Lattice & Arrow Diagram form of representations Example : If A = {1,2} , B = {3,4} ,then A B =(1,3) , (1,4) , (2,3) , (2,4) } , then its graphical representation is given below . Lattice form Arrow Diagram X

Domain and Range of the relation: Let R be a relation from a set A to the set of all first components or co-ordinates of the ordered pairs belonging to R is called the domain of R, While the set of all second components or co-ordinates of the ordered pairs belonging to R is called the range of R. Thus, Dom (R) = { a : (a,b) R } and Range (R) = { b : (a,b) R } Example : In the relation R = {(1,3),(2,4)} , DomR= {1,2} and Range R = {3,4}

R R = R2= {(x,y) : x , y R } represents whole plane . R R R = R3(3D) = {(x,y,z) : x , y, z R } represents whole space. Example : Example: R R = R2(2D)

RELATIONS Definition: Let A and B be two non empty sets, then each subset of A B defines a relation R from A to B . Let R be a subset of A B and (a,b) R then we say a is related to b by the relation R i.e. a R b. Here B is co-domain . Example : Let A = {1,2} , B = {3,4} . Here A B = { (1,3),(1,4),(2,3),(2,4)} R = {(1,3),(2,4)} is a relation but R = {(1,2),(2,4)} is not a relation as R is not a subset of A B .

Explanation by teacher https://drive.google.com/file/d/1yxHpjV- P79fv0PvGxne012HmHSVYF6jl/view?usp=sharing

Visusal Explanation https://youtu.be/HXccn91fAx8

Inverse Relation: Let A and B be two sets and let R be a relation from a set A to a set B. Then the inverse of R, denoted by R-1, is a relation from B to A and is defined defined by R-1 = {(b,a):(a,b) R}. Example: Let A = {a,b,c} and B = {1,2,3} and R= {(a,1),(a,3),(b,3),(c,3)} then R-1= {(1,a),(3,a),(3,b),(3,c)} Remark : If a set A has m number of elements and that of B has n number of elements, then number of relations from A to B is 2mn Representation of a relation : 1. Tabular /Roster form : In this form , relation is represented by the set of all ordered pairs belonging to R .

Example : Define a relation on the set N of natural numbers by R = {(x,y) : y = x+5 ; x is a natural number less than 4 and x, y N } .Depict the relation using its domain and range . Ans : R ={((1,6),(2,7),(3,8) } Domain = {1,2,3} , Range = {6,7,8} Example : Let A = {1,2,3,...,14}. Define a relation R A to A by R = {(x,y) : 3x - y = 0 ; x,y A} , find its domain , co-domain and range. Ans : R = {(x,y) : 3x - y = 0 ; x,y A} R = {(1,3) , (2,6),(3,9),(4,12),(5,15)} Domain = {1,2,3,4,5} , Range = {6,9,12,15} , Co-domain = A Roster form, write from

CONCEPTS MAP Cartesian Product of two sets * Cartesian product: A B = {(a,b) : a A , b B} . * Cartesian product is not commutative . * n(A) = P , n(B) = q n(A B) = p.q * (a,b) = (x,y) a = x and b = y Relation * R is a relation from A to B (where A , B ) if R is a subset of A B = { (a,b) : a A , b B } * Domain R = { a : (a,b) R } * Range R = { b : (a,b) R }

DAV PUBLIC SCHOOL, POKHARIPUT BHUBANESWAR 20 SUBJECT : MATHEMATICS CHAPTER -RELATION CLASS-XI WORKSHEET - 1 Mark the correct alternative in each of the following (MCQ type) Each carry One mark 1. Let R be a relation on N defined by x + 2y = 8 .The domain of R is (a){2,4,8} (b){2,4,6,8} (c){2,4,6} 2. If two sets A & B are having 99 elements in common, then the no. of elements common to each of the sets A B & B A are (a) 299 (b) 992 (c) 100 Answe the following questios in one line (very Short type) Each carry One mark 3. Let A = { 9 , 10 , 11 , 12 , 13 } and let f : A N be defined by f(n) = the highest prime factor of n. Find the range of f . 4. Find x and y if (x+3,5) = (6,2x+y). Fill in the blanks of the following questions Each carry One mark 5. If set A has 3 elements and the set B = { 1,3,4,5} , the number of elements in A B is .... 6. If A = {1,2,3} , B = {1,3,5} such that R = {(1,1) , (2,5)} then R-1 is ... Solve the following questios (Short type) Each question carry 2 marks. 7. A = {1,2,3,5} and B = {4,6,9}.Define a relation R from A to B by R = { (x,y) : x , y A and difference between x and y is odd }.Write R in roste form. 8. The relation R is defined as R = {(x,x+5): x {0,1,2,3,4,5}} .Write R in roster form.Write its domain and range . (d) {1,2,3,4} d) 18

9. Define a relation R on the set N of natural numbers by R = {(x,y):y=x+5 , x is a natural number less than 4 ; x, y N} .Depict the relationship using roster form. Solve the following questios (Long type- I) Each question carry 4 mark 10. If A ={ 2,4,6,9 } and B = { 4,6,18,27,54 } , Let R be a relation from set A to set B defined as R = {(a,b) : a < b and a is a factor of b } .Find R . Also find domain and range. 11. Let A = { 1,2,3,...,20 } .Define a relation R from A to A by R = {(a,b): a - 2b = 0,a,b A} Depict the relation using roster form.Write its domain and range. Mark the correct alternative in each of the following (MCQ type) Each carry 1 mark 1. If n(A) = 4, n (B) = 3, n(A B C) = 24, then n(C) = (a) 288 (b) 1 (c) 12 2. The number of elements in the set {(a,b) : 2 a2 + 3 b2 = 35 ; a, b Z} (a) 2 (b) 4 (c) 8 Answe the following questios in one line (very Short type) Each carry 1 mark 3. If A B = {(a,x),(a,y),(b,x),(b,y)}.Find A and B . 4. If A = {1,2} , form the set A A A . Fill in the blanks of the following questions Each carry 1 mark 5. If n(A) = 2 , n(B) = 3 , the number of non -empty relations from A to B is .... 6. Let R be a relation defined by R = {(4,5),(1,4),(4,6),(7,6),(3,7) , then RoR is ... Solve the following questios (Short type) Each carry 1 mark 7. A = {1,2,3,5} and B = {4,6,9}.Define a relation R from A to B by R = { (x,y) : x , y A and difference between x and y is odd }.Write R in roste form. WORKSHEET - 2 (d) 2 (d) 12

8. Let A = {1,3,5} . Define a relation R from A to A as R = {(x,y): x+y > 6 ; x, y A}. Write R in roster form.Write its domain and range. 9. If R = {(x,y) : x,y Z , x2 + y2 4 } is a relation on Z , then find domain of R . Solve the following questios (Long type- I)-Each question carry 4 mark. 10. Let R be a relation defined on A = {1,2,3,4,5,6} as R = {(x,y) : y = x + (6/x) ; x ,y A } (i)Find the domain and range of R (ii)Draw an arrow diagram of the relation R. 11. Let A = { x : x = 3n , n 6 , n N}.Define a relation R from A to B by R = {(x,y): y = 2x ; x,y A}. a) Write R in roster form b)Write domain,range , co-domain c)Draw the arrow diagram for R. ................................

WORKSHEET - 3 Mark the correct alternative in each of the following (MCQ type)-Each carry 1 mark. 1. If A is the set of even natural numbers less than 8 and B is the set of prime numbers less than 7,then the number of relations from A to B is a) 99 b) 92 c) 32 2. If A = {x:x2-5x + 6 = 0}, B={2, 4}, C = {4, 5} then A x (B C) is (a) {(2, 4), (3, 4)} (b) {(4, 2), (4,3)} c) {(2,4), (3,4),(4,4}) d) {(2,,2), (3,3), (4,4), (5,5)} Answe the following questios in one line (very Short type)- Each carry 1 mark 3. Let A and B be two sets containing 2 and 4 elements respectively.Find the number of relations from A to B containing 3 or more elements . 4. Let Z be the set of integers .If A = {x Z : 2(x+2)(x2-5x+6) = 1} and B = { x Z: - 3 < 2x 1 < 9 } , then find number of relations from A to B . Fill in the blanks of the following questions - each carry 1-mark 5. If R = {(2,1).(4,7),(1,-2),...} , then the relation between the components of ordered pair is.. 6. Let R = {(x,y) : x,y Z , y = 2x 4 }. If (a,-2) and (4, b2) R , then values of a and b are ... Solve the following questios (Short type) Each carry two mark. 7. Let R be a relation from N to N defined by R = {(a,b) : a , b N and a = b + 5 }. Does (a,b) R , (b,c) R imply (a,c) R .Justify 8. Determine the domain and range of the relation R , where R = {(x,x3) : x is a prime number less than 10}. d)29

9. If A = { 2,3,4,5,6,7,8,9 }.Let R be a relation on A defined by R = { (x,y) : x , y A and x divides y }. Draw arrow diagram of R and R in roster form . Solve the following questios (Long type- I) Each carry 4 mark . 10. For any three sets A , B , C , prove that A (B C) = (A B) (A C) . 11. Let R be a relation on Q defined by R = {(a,b) : a , b Q and a b Z}.Show that (i) (a,b) R for all a , b Q (ii) (a,b) R (b,a) ) R (iii) (a,b) , (b,c) R (a,c) ) R

Relations

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