Relation and it's type Discrete Mathematics.pdf
- 1. SHRIRAMSWAROOP MEMORIAL UNIVERSITY DISCRETE MATHEMATICS (MMA1010) MCA – 1st Semester TOPIC: Relation and its types SUBMITTED TO: SUBMITTED BY: Dr. Ajay Singh Mohd. Shahil Alam 202310101050006
- 2. RELATION Relation in Mathematics is defined as the relationship between two sets. If we are given two sets set A and set B and set A has a relation with set B then each value of set A is related to a value of set B through some unique relation. Here, set A is called the domain of the relation, and set B is called the range of the relation. For example if we are given two sets, Set A = {1, 2, 3, 4} and Set B = {1, 4, 9, 16} then the ordered pair {(1, 1), (2, 4), (3, 9), (4, 16)} represents the relation defined as, R, A: → B {(x, y): y = x2: y ϵ B, x ϵ A}.
- 3. TYPES OF RELATIONS Empty relation: A relation will be known as empty relation if the elements of set have no relation with each other. Example:- R = Ø Universal Relation: In this relation, every element of set A has a relation with every element of set B. That's why this relation is also called a universal relation, and the universal relation and empty relation are also known as the trivial relation.
- 4. Identity Relation: In the identity relation, every element of A has a relation to itself only. The identity relation is described as follows: A : A → A Inverse Relation: Assume that there are two sets, set A and set B, and they have a relation from A to B. This relation will be described as R∈ A × B. The inverse relation will be obtained when we replace the first element of each pair with the second element in a set. The inverse relation is described as follows: R -1 = {(b, a) : (a, b) ∈ R}
- 5. Reflexive Relation: A relation will be known as reflexive relative if every element of set A is related to itself. The word reflexive means that in a set, the image of every element has its own reflection. Symmetric Relation: Suppose there are two elements in a set A, i.e., a, b. The relation R on a set A will be called symmetric relation if element 'a' has relation with 'b' is true, then 'b' has a relation with 'a' will also be true. The symmetric relation is described as follows: If (a, b) ∈ R then (b, a) ∈ R, for all a and b ∈ A
- 6. Transitive Relation: Suppose there are three elements in a set A, i.e., a, b, and c. The relation R on a set A will be known as transitive relation if 'a' has a relation with 'b' and 'b' has a relation with 'c', then 'a' will be also has a relation with 'c'. The transitive relation is described as follows: If (a, b) ∈ R , (b, c) ∈ R , then (a, c) ∈ R for all a, b, c ∈ A Equivalence Relation: A relation will be known as equivalence relation if it satisfies the properties of symmetric, transitive, and reflexive. Note: All relations are not functions, but all functions are relations.