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Properties of Relations

A relation from a set A to set B is a rule that assigns us the relation between elements of A to elements of B.
A relation can be between one set or it can be between many sets.

Relations:
Let X and Y are two sets. A relation, R, from X to Y is a subset of the Cartesian product X × Y, where
a Cartesian product of X × Y is the superset of any relation between X and Y. This is because a Cartesian product contains all the possible relations between X and Y.

The statement (x, y) $\epsilon$ R is read as x is related to y by the relation R, and is denoted by xRy or R(x, y).
Consider the two sets as X = {1, 2, 3} and Y = {4, 5, 6}
Then R = {(1, 4), (1, 5), (1, 6), (2, 4), (2, 5), (2, 6), (3, 4), (3, 5), (3, 6)} and will be graphically represented by its relation as:

Properties of Relations

The total number of possible relations between two sets can be found out as follows:
If X have a number of elements and Y have b number of elements, then the total number of possible relations between set X and Y will be 2ab.

Properties of Relations:
There are three important properties of Relations which are as follows:
Reflexive Property: A relation in which all the elements follow the property A$\rightarrow $A. i.e.
All the elements are related to themselves is known as reflexive relation.
If for all x in set X, the relation G = xRx holds true then G is said to be reflexive otherwise it would be irreflexive.

Symmetric Property: A relation in which all the elements follow the property such that, if A$\rightarrow $B then B$\rightarrow $A is said to be Symmetric relation.
If for all x and y in X, the relation G = xRy = yRx holds true then G is said to be Symmetric otherwise it would be antisymmetric.

Transitive Property: A relation in which all the elements follow the property such that, if A$\rightarrow $B and B$\rightarrow $C then A$\rightarrow $C is said to be Transitive relation.
It is possible that a relation may not have any one of the above mentioned properties, it may have some of these properties or it might agree to all the properties.
A relation on a set A is called an equivalence relation if it has all the above 3 properties, that is, if its reflexive, symmetric and transitive.

Solved Examples:
Example 1: Let B is a set equals to [11, 12, 13, 4, 5] and R be the relation as [(11, 11), (11, 12), (11, 13), (11, 4), (11, 5), (12, 12), (12, 4), (12, 5), (13, 13), (13, 4), (13, 5), (4, 4), (4, 5), (5, 5)]. Find if the relation is reflexive, symmetric and antisymmetric.
Solution: The relation R is reflexive, as (11, 11), (12, 12), (13, 13), (4, 4) and (5, 5) are all in R.
R is not symmetric because (11, 12) $\epsilon $ R but (12, 11) $\not{\epsilon}$ R.
R is antisymmetric as there is no pair (x, y) in R for which (y, x) is also in R, except for (11, 11), (12, 12), (13, 13), (4, 4) and (5, 5).

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