# Discrete Mathematics - 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 $x \times y$. If the ordered pair of G is reversed, the relation also changes.

Generally an n-ary relation R between sets $A_1, \dots ,\ and\ A_n$ is a subset of the n-ary product $A_1 \times \dots \times A_n$. The minimum cardinality of a relation R is Zero and maximum is $n^2$ in this case.

A binary relation R on a single set A is a subset of $A \times A$.

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 $\lbrace x \:| \: (x, y) \in R \:for\: some\: y\: in\: B \rbrace$The

**range**of R, Ran(R), is the set $\lbrace y\: |\: (x, y) \in R \:for\: some\: x\: in\: A\rbrace$

### Examples

Let, $A = \lbrace 1, 2, 9 \rbrace $ and $ B = \lbrace 1, 3, 7 \rbrace$

Case 1 − If relation R is 'equal to' then $R = \lbrace (1, 1), (3, 3) \rbrace$

Dom(R) = $\lbrace 1, 3 \rbrace , Ran(R) = \lbrace 1, 3 \rbrace$

Case 2 − If relation R is 'less than' then $R = \lbrace (1, 3), (1, 7), (2, 3), (2, 7) \rbrace$

Dom(R) = $\lbrace 1, 2 \rbrace , Ran(R) = \lbrace 3, 7 \rbrace$

Case 3 − If relation R is 'greater than' then $R = \lbrace (2, 1), (9, 1), (9, 3), (9, 7) \rbrace$

Dom(R) = $\lbrace 2, 9 \rbrace , Ran(R) = \lbrace 1, 3, 7 \rbrace$

## Representation of Relations using Graph

A relation can be represented using a directed graph.

The number of vertices in the graph is equal to the number of elements in the set from which the relation has been defined. For each ordered pair (x, y) in the relation R, there will be a directed edge from the vertex ‘x’ to vertex ‘y’. If there is an ordered pair (x, x), there will be self- loop on vertex ‘x’.

Suppose, there is a relation $R = \lbrace (1, 1), (1,2), (3, 2) \rbrace$ on set $S = \lbrace 1, 2, 3 \rbrace$, it can be represented by the following graph −

## Types of Relations

The

**Empty Relation**between sets X and Y, or on E, is the empty set $\emptyset$The

**Full Relation**between sets X and Y is the set $X \times Y$The

**Identity Relation**on set X is the set $\lbrace (x, x) | x \in X \rbrace$The Inverse Relation R' of a relation R is defined as − $R' = \lbrace (b, a) | (a, b) \in R \rbrace$

**Example**− If $R = \lbrace (1, 2), (2, 3) \rbrace$ then $R' $ will be $\lbrace (2, 1), (3, 2) \rbrace$A relation R on set A is called

**Reflexive**if $\forall a \in A$ is related to a (aRa holds)**Example**− The relation $R = \lbrace (a, a), (b, b) \rbrace$ on set $X = \lbrace a, b \rbrace$ is reflexive.A relation R on set A is called

**Irreflexive**if no $a \in A$ is related to a (aRa does not hold).**Example**− The relation $R = \lbrace (a, b), (b, a) \rbrace$ on set $X = \lbrace a, b \rbrace$ is irreflexive.A relation R on set A is called

**Symmetric**if $xRy$ implies $yRx$, $\forall x \in A$ and $\forall y \in A$.**Example**− The relation $R = \lbrace (1, 2), (2, 1), (3, 2), (2, 3) \rbrace$ on set $A = \lbrace 1, 2, 3 \rbrace$ is symmetric.A relation R on set A is called

**Anti-Symmetric**if $xRy$ and $yRx$ implies $x = y \: \forall x \in A$ and $\forall y \in A$.**Example**− The relation $R = \lbrace (x, y)\to N |\:x \leq y \rbrace$ is anti-symmetric since $x \leq y$ and $y \leq x$ implies $x = y$.A relation R on set A is called

**Transitive**if $xRy$ and $yRz$ implies $xRz, \forall x,y,z \in A$.**Example**− The relation $R = \lbrace (1, 2), (2, 3), (1, 3) \rbrace$ on set $A = \lbrace 1, 2, 3 \rbrace$ is transitive.A relation is an

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