Types of Relations: Reflexive, Symmetric, Transitive A relation R on a set A is a subset of the Cartesian product of A with itself, meaning it's a set of al...
Types of Relations: Reflexive, Symmetric, Transitive
A relation R on a set A is a subset of the Cartesian product of A with itself, meaning it's a set of all ordered pairs (a, b) such that a relates to b.
Reflexive Relation:
A relation R on a set A is reflexive if for every element a in A, the relation R(a, a) is true. In other words, a has the same relation to itself as any other element in the set.
Example:
Consider the relation R on the set of all people in a city. R(a, b) if a and b are friends. Clearly, this relation is reflexive because for every person in the city, they are friends with themselves.
Symmetric Relation:
A relation R on a set A is symmetric if for every element a in A, if R(a, b) is true, then R(b, a) is also true. In other words, if a and b are related, then b and a must also be related.
Example:
Consider the relation R on the set of all students in a school. R(a, b) if a and b are in the same class. This relation is symmetric because for every student in the school, if they are in the same class, then they are also in the same class with the other student.
Transitive Relation:
A relation R on a set A is transitive if for every elements a, b, and c in A, if R(a, b) and R(b, c), then R(a, c) is also true. In other words, if a is related to b, and b is related to c, then a is also related to c.
Example:
Consider the relation R on the set of all students in a school. R(a, b) if a is the headmaster and b is the vice-president. Similarly, R(b, a) is also true because b is the headmaster and a is the vice-president. This relation is transitive because for every student in the school, if they are the headmaster or the vice-president, then they are also the principal.
In summary, a relation is reflexive if a relates to itself for every element, symmetric if b relates to a if a relates to b, and transitive if a relates to b and b relates to c, then a relates to c