Relation Formula Sheet

A relation describes the connection between elements of two sets. In functions, each element of one set is associated with exactly one element of another set.

Neetesh Kumar | May 29, 2024 $\space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space$ Share this Page on:

• 1. Definition
• 2. Types of Relations

1. Definition:

Let $A$ and $B$ be two sets. Then, a relation $R$ from $A$ to $B$ is a subset of $A \times B$.
Thus, $R$ is a relation from $A$ to $B \iff R \subseteq A \times B$.

Total Number of Relations: Let $A$ and $B$ be two non-empty finite sets consisting of $m$ and $n$ elements respectively. Then, $A \times B$ consists of $mn$ ordered pairs. So the total number of subsets of $A \times B$ is $2^{mn}$.

Domain and Range of a Relation: Let $R$ be a relation from a set $A$ to a set $B$. Then, the set of all first components or coordinates of the ordered pairs belonging to $R$ is called the domain of $R$, while the set of all second components or coordinates of the ordered pairs in $R$ is called the range of $R$.

Thus, $\text{Domain}(R) = \{a : (a, b) \in R\}$ and, $\text{Range}(R) = \{b : (a, b) \in R\}$

It is evident from the definition that the domain of a relation from $A$ to $B$ is a subset of $A$, and its range is a subset of $B$.

Inverse Relation: Let $A$, $B$ be two sets and let $R$ be a relation from a set $A$ to a set $B$.
Then the inverse of $R$, denoted by $R^{-1}$, is a relation from $B$ to $A$ and is defined by $R^{-1} = \{(b, a) : (a, b) \in R\}$ Clearly, $(a, b) \in R \iff (b, a) \in R^{-1}$ Also,
$\text{Domain}(R) = \text{Range}(R^{-1}) \quad \text{and} \quad \text{Range}(R) = \text{Domain}(R^{-1})$

Note: Relation on a set: If $R$ is a relation from set $A$ to $A$ itself then $R$ is called a relation on set $A$.

2. Types of Relations:

In this section, we intend to define various types of relations on a given set $A$.

• Void Relation: Let $A$ be a set. Then $\emptyset \subseteq A \times A$ is a relation on $A$. This relation is called the void or empty relation on $A$.

• Universal Relation: Let $A$ be a set. Then $A \times A \subseteq A \times A$ is a relation on $A$. This relation is called the universal relation on $A$.

• Identity Relation: Let $A$ be a set. Then the relation $I_A = \{(a, a) : a \in A\}$ on $A$ is called the identity relation on $A$. In other words, a relation $I_A$ on $A$ is called the identity relation if every element of $A$ is related to itself only.

• Reflexive Relation: A relation $R$ on a set $A$ is said to be reflexive if every element of $A$ is related to itself. Thus, $R$ on a set $A$ is not reflexive if an element $a \in A$ exists such as $(a, a) \notin R$. Every identity relation is reflexive but every reflexive relation is not identity.

• Symmetric Relation: A relation $R$ on a set $A$ is said to be a symmetric relation iff $(a, b) \in R \iff (b, a) \in R$ i.e.,$a R b \iff b R a$

• Transitive Relation: Let $A$ be any set. A relation $R$ on $A$ is said to be a transitive relation iff $(a, b) \in R \quad \text{and} \quad (b, c) \in R \implies (a, c) \in R$ i.e., $a R b \quad \text{and} \quad b R c \implies a R c$

• Antisymmetric Relation: Let $A$ be any set. A relation $R$ on set $A$ is said to be an antisymmetric relation iff $(a, b) \in R \quad \text{and} \quad (b, a) \in R \implies a = b$

• Equivalence Relation: A relation $R$ on a set $A$ is said to be an equivalence relation on $A$ iff It is reflexive i.e. $(a, a) \in R$ for all $a \in A$
  It is symmetric i.e. $(a, b) \in R \implies (b, a) \in R$
  It is transitive i.e. $(a, b) \in R \quad \text{and} \quad (b, c) \in R \implies (a, c) \in R$
  

Not every relation which is symmetric and transitive needs to be also reflexive.

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