Discrete Mathematical Structures - Relations

Whenever sets are being discussed, the relationship between the elements of the sets is the next thing that comes up. Relations may exist between objects of the same set or between objects of two or more sets.

Definition and Properties
A binary relation R from set x to y (written as xRy or R(x,y)) is a subset of the Cartesian product x×y . If the ordered pair of G is reversed, the relation also changes.
Generally an n-ary relation R between sets A1,…, and An is a subset of the n-ary product A1×⋯×An. The minimum cardinality of a relation R is Zero and maximum is n2 in this case. A binary relation R on a single set A is a subset of A×A .
For two distinct sets, A and B, having cardinalities m and n respectively, the maximum cardinality of a relation R from A to B is mn.

Domain and Range
If there are two sets A and B, and relation R have order pair (x, y), then −
    The domain of R, Dom(R), is the set {x|(x,y)∈R for some y inB}
    The range of R, Ran(R), is the set {y|(x,y)∈R for some x in A}

Examples
Let, A={1,2,9}
and B={1,3,7}
Case 1 − If relation R is 'equal to' then R={(1,1),(3,3)}
Dom(R) = {1,3},Ran(R)={1,3}
Case 2 − If relation R is 'less than' then R={(1,3),(1,7),(2,3),(2,7)}
Dom(R) = {1,2},Ran(R)={3,7}
Case 3 − If relation R is 'greater than' then R={(2,1),(9,1),(9,3),(9,7)}
Dom(R) = {2,9},Ran(R)={1,3,7}

Types of Relations
*The Empty Relation between sets X and Y, or on E, is the empty set ∅

*The Full Relation between sets X and Y is the set X×Y

*The Identity Relation on set X is the set {(x,x)|x∈X}

*The Inverse Relation R' of a relation R is defined as − R′={(b,a)|(a,b)∈R}
Example − If R={(1,2),(2,3)} then R′ will be {(2,1),(3,2)}

*A relation R on set A is called Reflexive if ∀a∈A is related to a (aRa holds)
Example − The relation R={(a,a),(b,b)} on set X={a,b} is reflexive.

*A relation R on set A is called Irreflexive if no a∈A is related to a (aRa does not hold).
Example − The relation R={(a,b),(b,a)} on set X={a,b} is irreflexive.

*A relation R on set A is called Symmetric if xRy implies yRx, ∀x∈A and ∀y∈A .
Example − The relation R={(1,2),(2,1),(3,2),(2,3)} on set A={1,2,3} is symmetric.

*A relation R on set A is called Anti-Symmetric if xRy and yRx implies x=y∀x∈A and ∀y∈A.
Example − The relation R={(x,y)→N|x≤y} is anti-symmetric since x≤y and y≤x implies x=y.

*A relation R on set A is called Transitive if xRy and yRz implies xRz,∀x,y,z∈A.
Example − The relation R={(1,2),(2,3),(1,3)} on set A={1,2,3} is transitive.

*A relation is an Equivalence Relation if it is reflexive, symmetric, and transitive.
Example − The relation R={(1,1),(2,2),(3,3),(1,2),(2,1),(2,3),(3,2),(1,3),(3,1)} on set A={1,2,3} is an equivalence relation since it is reflexive, symmetric, and transitive.