- Symmetric relation
A relation R, on a set A is said to be symmetric, if aRb and bRa, for all
a, b ∈ A.
In symbols, aRb implies bRa
Eg. :In the set of all lines let R denote the relation “is parallel to”. If a line l 1 is
parallel to another line l 2 , then l 2 is parallel to l 1. Hence l 1 Rl 2 ⇒ l 2 Rl 1. So R
is symmetric.
- Transitive relation
A relation R on a set A is said to be transitive if aRb, bRc imply aRc for all a,b,c ∈ A.
In symbols, aRb, bRc imply aRc
Eg. :Consider the set N of natural numbers. Let a, b, c ∈ N. Let r denote the relation
“is less than”. R is transitive since a < b and b < c ⇒ a < c.
- Equivalence relation
A relation R on a set A is called an equivalence relation if it is reflexive, symmetric
and transitive. In other words, a relation in a set A is equivalence relation if the
following conditions are satisfied.
a R a, for all a ∈ A
a R b implies b R a, for all a, b ∈ A
a R b and b R c imply a R c, for all a, b, c ∈ A
Examples from family may also used.
Assume x, y, and z are brothers from the same family A. X∈A, XRX, XRY implies YRZ
XRY and YRZ imply XRZ for all X, Y, Z ∈ A
Eg. :In the set of lines on a plane, the relation “is parallel to” is reflexive, symmetric
and transitive. Therefore the relation “is parallel to” is an equivalence relation.
