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RELATIONS Plan •Definition with various examples based on our daily life. •Presentation with diagrams. •Presentation with examples. •Idea of its different types. •Focusing on solution of assignments. 1 Ordered Pairs Let Mrs. Rama has three suits A= {S1, S2, S3} and two necklaces B= {N1, N2}. She is going to attend a reception party with her husband Ramesh. She has to find the best match of suits and necklaces i.e. she has to make pair of suits and necklaces in the following manner: (S1,N1),(S1,N2),(S2,N1),(S2,N2),(S3,N1) and (S3,N2) Thus 6 distinct ordered pairs can be constructed. 2 •Discussion of previous knowledge required to define a relation. Ordered Pairs: Let us consider the two sets say A and B such as A= {a, b} and B={p, q}. Let us form pairs of elements taking one from each set. Then pairs are (a , p) ,(a, q),(b, p),(b, q). We call such pairs as ordered pairs. Thus we can define an ordered pair as follows: 3 Ordered pairs An ordered pair is a pair of elements in specified order separated by a comma and enclosed within parentheses (small brackets). For example (a) (a,2), (b,3), (c,4), (d,5) are ordered pairs whose first components a, b, c and d are English alphabets taken in order and second components 2,3,4 and 5 are natural numbers. 4 (b) Points in a plane are also represented by ordered pairs (x, y) for abscissa’s ‘x’ and ordinates ‘y’. Equality of Ordered Pairs: Two ordered pairs (x, y) and (m, n) are said equal if x=m and y=n. Cartesian Product of two sets: Let us consider two sets A={Ram, Shyam}, B={Sita, Radha}. Then possible ordered pairs are (Ram, Sita), (Ram, Radha), (Shyam, Sita), (Shyam, Radha)on 5 considering first components from set A and second components from set B. The set of all these ordered pairs is called the Cartesian product of set A and B and is written as AXB={(Ram, Sita), (Ram, Radha), (Shyam, Sita), (Shyam, Radha)}. Thus we can define Cartesian product as follows: 6 Definition The Cartesian product of two non-empty sets is the set of all ordered pairs with first components from first set and second components from second set. Symbolically the Cartesian product of A and B is If either of A and B is the null set then A X B is null set. 7 Cardinal number of Cartesian product of sets Let A and B be finite sets then the number of all the ordered pairs of A X B is called Cardinal number of A X B . We write Cardinal of a set A as n (A). n (A X B) = n(A).n(B) And n(B X A) = n(B).n(A) 8 For example: A = {1, 2}, B = {a, b} then A X B = {(1, a), (1, b), (2, a), (2, b)} n (A X B) = 4 and n(A)=2, n(B)=2 n (A X B) = 4 and n(A)=2, n(B)=2 n (A X B) = n(A) . n (B) 9 Relation: Relation is familiar in our daily life such as husband and wife, father and son, brother and sister, mother and daughter etc. In Mathematics we observe ‘a line l is perpendicular to line m’, ‘a line l is parallel to line m’, ∆PQR is similar to ∆ XYZ’, ‘∆ ABC is congruent to ∆LMN’, ‘‘ Set A is equal to Set B’, ‘Set A is subset of Set B’ etc. In these relations we see that a pair of objects is involved in specific order. Let us consider two sets and introduce a relation between the objects from first set to second set. For example: (1) A={Ram, Bharat} , B={Dashrath } . There is a relation ‘ is son of ‘ between the elements of set A and set B. If we denote R for the relation ‘ is son of’ then we have Ram R Dashrath, 10 Example-2 : Let us consider two sets A={Patna, Kolkatta, Ranchi}, B={Bihar, West Bengal, Jharkhand, Madhya Pradesh}. We define a relation ‘is capital of’ then the relation can be represented as follows: Figure- 1 A B 11 12 R = {(2,4), (2,6),(2,10),(3,6),(3,9)} Obviously A X B, . With the help of examples we define a relation as follows: Relation R from Definition: Let A and B be any two non empty sets. Then relation R from set A to set B is a subset of A × B. Thus a relation is a set of ordered pairs. Of course, this means that every graph is relation. 13 Examples of relations: Example : (1) R= {(2,1), (1,3), (3,2),(4,3)} with corresponding graph in figure Example : (2) R= {(0,0), (1,1), (2,2),(3,3), (4,4)} with corresponding graph in figure 14 Example : (2) R= {(0,0), (1,1), (2,2),(3,3), (4,4)} with corresponding graph in figure Figure - 4 15 Binary Relation on a set : If A be a non-empty set, then a subset of A X A is called a binary relation or simply a relation on A. Example: Let N be the set of natural numbers. There is a relation ‘ has as its square’ from the set N to N. If we write R for the relation ‘has as its square’, then 1R1, 2R4, 3R9, 4R16, 5R25, ………………… In the form of ordered pairs we write R = {(1,1), (2,4), (3,9), (4,16), (5,25), ………..} In a diagram we represent it as 16 N N 17 Domain, Co - domain and Range of a Relation : Let R be a relation from set A to set B , then the domain of the relation R, denoted by D, is the set of all first components of the ordered pairs of R The range of R the set of all second components of the ordered pairs of R i.e. E = {y: y } If R is a relation from set A to set B then set B is called co-domain of R. 18 Example: (1) Let A = {Delhi, Kabul, Kathmandu} B = {India, Sri Lanka, Afghanistan, Nepal, Pakistan} We define a relation R as ‘is capital of’, then R = { (Delhi, India), (Kabul, Afghanistan), (Katmandu, Nepal)}. Then domain of R is D = {Delhi, Kabul, Katmandu} Range E of R is E = {India, Afghanistan, Nepal} And Co-domain of R is {India, Sri Lanka, Afganistán, Nepal, Pakistán} 19 Example: (2) Let A = {1, 2, 3 ,4} B = { 1,2, 3, } R is defined with the graph as follows: i.e, R = { (1, 3), (2, 1), (3, 2) , (4,3)} Then D = { 1, 2, 3,4} is domain of R E = {1, 2, 3} is range of R And set B = { 1 , 2, 3 } is co-domain of R 20 Example (3): Let us consider a relation R between the sets X and Y follows: The relation R can be written in roster form as R = {(5, 3), (6, 4), (7, 5)} In set builder form, the relation R can be expressed as Then domain ‘D’ of R is, D = { 5, 6, 7 } Range E of R is, E = {3, 4, 5 } And set Y = { 3, 4, 5 } is co-domain of R. 21 Figure – 8 From this family tree we see that Ramesh is son of Rakesh and Reshma . Rimjhim is daughter of Radha and Ramesh. Let us define a relation R as the set of ordered pairs (x, y) where x = a daughter and y = the mother of x then the relation is 22 R = { (Kajal, Reshma), (Rimjhim, Radha), (Kritika, Kajal) } Then domain of R is D = {Kajal, Rimjhim, Kritika } Range of R is E = { Reshma , Radha, Kajal }. If we define a relation R , where x = a brother and y = the sister of x. Then this relation is R = {(Ramesh, Kajal ),(Ravi, Rimjhim) } 23 Thus domain D of R is D = { Ramesh , Ravi } and the range E of R is E = { Kajal, Rimjhim } In this way many examples can be produced for Domain, Co-domain and Range of a relation R. Points to remember 24 Some Particular types of Relations Empty relation: If A be a non-empty set called void relation or empty relation. i.e. no element of A is related to any element of A. R is an empty relation in A. Universal Relation: A relation R in a set A is called universal relation, if each element of A is related to every element of A. 25 26 Example : Let T be the set of all triangles in a plane. A relation R on T defined by R = { (T1, T2) : T1 is congruent to T2} , R is reflexive as every triangle is congruent to itself. 27 Example (2): The identity relation on a non-empty set A is symmetric. Example (3) : The universal relation on a non-empty set A is symmetric. 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44