Tiêu đề RELATIONS & BINARY OPERATION (Q).pdf ✅

\\ Chapter – 1 DEFINITION A relation R, from a non-empty set A to another non-empty set B, is a subset of AB Equivalently, any subset of AB is relation from A to B. Thus, R is a relation from A to B  R  AB  R  a, b: a  A, b B Example: Let A  1, 2, B  a, b, c Let R  1, a, 1, c Here R is a subset of AB and hence it is a relation from A to B. 2.1 DOMAIN OF A RELATION Let R be a relation from A to B. The domain of relation R is the set of all those elements a  A such that a, bR for some b  B . Thus Dom (R) = a  A:a, bR for some b B Thus domain of R = set of first components of all the ordered pair which belong to R. 2.2 RANGE OF A RELATION Let R be a relation from A to B. The range of R is the set of all those elements b  R such that a, bR for some a  A . Thus range of R  b B :a, bR for some a A. Range of R = set of second components of all the ordered pairs which belong to R. Set B is called as codomain of relation R. Example1: Let A  2, 3, 5 and B  4, 7, 10, 8 Let aRb  a divides b Then Dom R  2, 5 and range of R  4, 10, 8 Codomain of R  B  4, 7, 10, 8 Example2: Let A  1, 2, 3, B  2, 4, 6, 8 Let R be a relation defined from A to B by xRy  y is double of x,  x  A Then 1R2, 2R4, 3R6  R  1, 2, 2, 4, 3, 6 A relation from a set A to set B can be represented in any one of the following four forms. 3.1 ROSTER FORM In this form a relation R is represented by the set of all ordered pairs belonging to R. Example: Let A = {–1, 1, 2} and B = {1, 4, 9, 10} Let aRb means a  b 2 RELATIONS 1 THEORY CONTENT OF RELATION & BINARY OPERATIONS DOMAIN AND RANGE OF A RELATION 2 REPRESENTATION OF A RELATION 3
Then R (in roaster form) = {(–1, 1), (1, 1), (2, 4)} 3.2 SET-BUILDER FORM In this form, the relation R is represented as a, b: a  A, b B, a.....b, the blank is to be replaced by the rule which associates a to b. Example: Let A = {1, 3, 5, 7}, B = {2, 4, 6, 8} Let R = {(1, 2), (3, 4), (5, 6), (7, 8)}, then R in the builder form can be written as R  a, b: a  A, b B; a  b  1 3.3 BY ARROW DIAGRAM In this form, the relation R is represented by drawing arrows from first component to the second component of all ordered pairs belonging to R. Example: Let A = {1, 2, 3, 4}, B = {0, 2, 4} and R be relation ‘is less than’ from A to B, then R  1, 2, 1, 4, 2, 4, 3, 4 This relation R from A to B can be represented by the arrow diagram as shown in the figure. 1 2 3 4 0 2 4 A B 3.4 BY LATTICE METHOD In this form, the relation R is represented by drawing dots in the lattice for all ordered pairs which satisfy the given relation R. Example: Let A = {1, 2, 3, 4}, B = {1, 2, 3, 4} and R be the relation ‘is a divisor of’ from A to B, then R  1, 1, 1, 2, 1, 3, 1, 4, 2, 2, 2, 4, 3, 3, 4, 4 This relation R from A to B can be represented by the lattice as shown in the given figure. The points marked by dots represent the ordered pairs satisfied by the given relation. x y 4 3 2 1 O 1 2 3 4 Let A and B be two non empty finite sets having p and q elements respectively. Then nAB  nA.nB  pq Therefore, total number of subsets of pq AB  2 Since each subset of AB is a relation from A to B, therefore total number of relations form A to B is pq 2 Example: Let A = {1, 2}, B = {3, 4, 5} Then nAB  nA.nB  23  6  Number of relations from A to B = 6 2 = 64 TOTSL NUMBER OF RELATIONS 4
Important formulae/points  If R is relation from A to B and (a, b)  R, then we also write a R b (read as a is not related to b)  aRb show that a is the domain and b is the range. 1. Empty relation: A relation R from A to B is called an empty relation or a void relation if R = . Example: Let A = {2, 4, 6} and B = {7, 11}. Let R = {(a, b) : a  A, b  B and a  b is even} Since none of the numbers 2–7, 2–11, 4–7, 4–11, 6–7, 6–11 is an even number, therefore R = . Hence R is an empty relation from A to B. 2. Universal relation: A relation R from A to B is said to be the universal relation if R  AB. Example: Let A = {1, 2}, B = {1, 3} Let R = {(1, 1), (1, 3), (2, 1), (2, 3)} Here R  AB Hence R is the universal relation from A to B. 1. Empty relation: A relation R on a set A is said to be an empty relation or a void relation if R = . Example: Let A = {1, 2, 3, 4, 5} Let R = {(a, b) : a – b = 12 and a, b  A}. We observe that a – b 12 for any two elements of A. Therefore, (a, b)  R for any a, b  A.  R does not contain any element of A × A  R is empty set  R is the empty relation (void relation) on A 2. Universal relation: A relation R on a set A is said to be universal relation on A if R  A A . Example: Let A = {1, 2}. Let R = {(1, 1), (1, 2), (2, 1), (2, 2)}. Here R  A A Hence R is the universal relation on A. 3. Identity relation: A relation R on a set A is said to be identity relation on A if R = a, b: a  A, b  A and a  b. Thus identity relation R = a, a:  a  A. Identity relation on set A is also denoted by A I . Example: Let A = {1, 2, 3, 4} Then IA  1, 1, 2, 2, 3, 3, 4, 4 Note: (i) Empty relation and universal relation are called trivial relations. (ii) In an identity relation on A, every element of A should be related to itself only. Illustration 1 Question: Let A = {3, 5}, B = {7, 11} Let R = a, b: a  A, b  B, a  b is even Show that R is an universal relation from A to B. Solution: Given, A  3, 5, B  7, 11 Now, R  a, b: a  A, b B and a  b is even = {(3, 7), (3, 11), (5, 7), (5, 11)} Also AB  3, 7, 3,11, 5, 7, 5,11 TYPES OF RELATIONS FROM ONE SET TO ANOTHER SET 5 TYPES OF RELATIONS ON A SET 6
Clearly, R  AB Hence R is an universal relation from A to B. 1. Reflexive relation: A relation R on a set A is said to be reflexive if a, aR, a  A. Example: Let A = {1, 2, 3}. Let R1  1, 1, 2, 2, 3, 3 R2  1, 1, 2, 2, 3, 3, 1, 2, 2, 1, 1, 3 R3  2, 2, 2, 3, 3, 2, 1, 1 Here R1 and R2 are reflexive relations on A. R3 is not a reflexive relation on A as (3, 3)  A . Note: The identity relation is always a reflexive relation but a reflexive relation may or may not be the identity relation. In the examples given above R1 is both reflexive and identity relation on A whereas R2 is a reflexive relation on A but not an identity relation on A. 2. Symmetric relation: A relation R on a set A is said to be a symmetric relation if a, bR  b, aR, where a, b  A . Example: Let A = {1, 2, 3}. Let R1  1, 2, 2, 1 R2  1, 2, 2, 1, 1, 3, 3, 1 Here R1 and R2 are symmetric relations on A. Let R3  2, 3, 3, 2, 2, 2 and R4  2, 3, 3, 1, 1, 3. Then R3 is a symmetric relation on A because     3 2 3 2, 3 R  3, R But R4 is not a symmetric relation on A because   3 4 2, R and   2 4 3, R 3. Transitive relation: A relation R on a set A is said to be a transitive relation if a, bR and b, cR  a, cR , where a, b, c  A . Example: Let A = {1, 2, 3} Let R1 = {1, 2), (2, 3), (1, 3), (3, 2)}. Let R2 = {(1, 3), (3, 2), (1, 2)}. Then R1 is not transitive relation on A because (2, 3)  R and (3, 2)  R but (2, 2)  R . Finally R2 is also a transitive relation. 4. Antisymmetric relation: A relation R on a set A is said to be antisymmetric if a, bR and b, aR  a = b. Thus R is antisymmetric if a  b , then both (a, b) and (b, a) cannot belong to R. One of them may belong to R. Example: Let A = {1, 2, 3}. Let R1  1, 2, 1, 3, 1, 1 R2  1, 2 R3  1, 2, 2, 1 Here R1 and R2 are antisymmetric relations on A but R3 is not antisymmetric relation on A. Note: A relation which is not symmetric is not necessarily antisymmetric. Example: Let A = {1, 2, 3}. Let R = {(1, 2), (2, 1), (2, 3)}. Here R is not symmetric as (2, 3)  R but (3, 2)  R Also R is not antisymmetric because (1, 2)  R and (2, 1)  R but 1  2 REFLEXIVE, SYMMETRIC AND TRANSITIVE REALTIONS 7 EQUIVALENCE REALATION ON A SET 8