Relations By Prof Nutan Patel Asst professor ITNU

Relations By Prof. Nutan Patel Asst. professor IT-NU. nutan. patel@nirmauni. ac. in 3 CA 1155 Mathematics Foundation

Relation • Introduction The word “Relation” is one of its common meaning indicates a tie between two objects. If we considering a set pf persons, then there are many types of family relationships that may or may not hold between the persons, such as father of, mother of, brother of, sister of, cousin of, uncle of, etc. The mathematical concept of a relation deals with the way the variables are related or paired. In mathematics, the expressions such as ‘is less than’, ‘is greater than’, ‘is square root of’, ’is cube of’ etc are relations. 3 CA 1155 Mathematics Foundation

• Binary Relation Let A and B be two non-empty sets then any subset of R of the Cartesian product A X B is called a binary relation R from A to B. R is a set of ordered pairs where each first element comes from A and each second element from B. Thus, we denote it with an ordered pair (a, b), where a A and b B. We also denote the relationship with a. Rb, which is read as ‘a related to b’. Example-1. Suppose we say that two countries are adjacent if they have some part of their boundaries common. Then, “is adjacent to”, is a relation R on the countries on the earth. Thus, we have, (India, Nepal) R, but (Japan, Sri Lanka) R. Example-2. Let A be the set of all possible inputs to a given computer program, and let B be the set of all possible outputs from the same program. Then the relation R from A to B can be defined as a. Rb, if and only if b is the output of the program when input a is used. 3 CA 1155 Mathematics Foundation

Example- 3. Let A = { 1, 2, 3} , B= { 1, 2, 3, 4} Relation R 1={(a, b) / a=b} = {(1, 1), (2, 2), (3, 3)}. R 2= {(a, b)/ a < b} = {(1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4)}. R 3={(a, b)/ a=2 b} = {(1, 2), (2, 4)} R 4={(a, b)/ a 2 + b 2 < 20} = {(1, 1), (1, 2), (1, 3), (1, 4), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)} 3 CA 1155 Mathematics Foundation

• Classification of relations: Let R be a relation on a set A. 1. Reflexive: R is said to be a reflexive relation on A, if a. Ra, for all a A. For example: i. R={(a, b)/ a=b} ii. R={(a, b)/ a ≥ b} iii. R={(a, b)/ a < b} iv. R “is adjacent to”, a relation adjacent countries if they have some part of their boundaries common. 2. Symmetric: R is said to be a symmetric relation on A, if a. Rb then b. Ra. For example: i. R={(a, b)/ a=b} i. R={(a, b)/ a ≥ b} ii. R={(a, b)/ a < b} iv. R “is adjacent to”, a relation adjacent countries if they have some part of their boundaries common. 3 CA 1155 Mathematics Foundation

3. Antisymmetric: R is said to be a antisymmetric, if a. Rb, b. Ra then a=b. For example: i. R={(a, b)/ a=b} ii. R={(a, b)/ a ≥ b} iii. R={(a, b)/ a ≤ b} iv. R “is adjacent to”, a relation adjacent countries if they have some part of their boundaries common. 4. Transitive: R is said to be a transitive, if a. Rb, b. Rc then a. Rc. For example: i. R={(a, b)/ a=b} ii. R={(a, b)/ a ≥ b} iii. R={(a, b)/ a < b} iv. R “is adjacent to”, a relation adjacent countries if they have some part of their boundaries common. 3 CA 1155 Mathematics Foundation

• Exercise- Classify the following Relation into Reflexive, symmetric, antisymmetric and transitive. 1. R={(a, b)/ a divides b} 2. R={(a, b)/ a is brother of b} 3. R={(a, b)/ a and b are both whats app user} 4. R={(A, B)/ A B} 5. R={(A, B)/ A B= } 3 CA 1155 Mathematics Foundation

• Equivalence Relation: A relation R on a set A is called an equivalence relation, If R is reflexive, symmetric and transitive. Example: 1. R={(a, b)/ a=b} 2. R={(a, b)/ a ≥ b} 3. R={(a, b)/ a ≤ b} 4. R={(a, b)/ a and b are both whats app user} 5. R={(a, b)/ a and b are in this class} 3 CA 1155 Mathematics Foundation