Types of Relations in Discrete Mathematics – Definitions, Examples & Solved Problems

Relations are a fundamental concept in Discrete Mathematics and Graph Theory. They are widely used in equivalence relations, partial order relations, posets, lattices, and many real-life applications. Questions based on relations are frequently asked in GTU and engineering mathematics examinations.

In this article, we explain the types of relations in a clear and exam-oriented manner with suitable examples and fully solved problems.

---

Question 1

Explain types of a relation with a suitable example.

Solution

Definition of Relation

Let A and B be two non-empty sets. A relation R from set A to set B is defined as a subset of the Cartesian product A × B.

Mathematically,

$$ R \subseteq A \times B $$

---

Types of Relations

1. Reflexive Relation

A relation R on a set A is said to be reflexive if every element of A is related to itself.

$$ (a,a) \in R \quad \text{for all } a \in A $$

Example:

Let $A = \{1,2,3\}$. Then the relation

$$ R = \{(1,1),(2,2),(3,3)\} $$

is a reflexive relation.

---

2. Symmetric Relation

A relation R is said to be symmetric if whenever $(a,b)$ belongs to R, then $(b,a)$ also belongs to R.

$$ (a,b) \in R \Rightarrow (b,a) \in R $$

Example:

$$ R = \{(1,2),(2,1),(3,3)\} $$

Hence, R is a symmetric relation.

---

3. Transitive Relation

A relation R is called transitive if whenever $(a,b)$ and $(b,c)$ belong to R, then $(a,c)$ must also belong to R.

$$ (a,b) \in R \text{ and } (b,c) \in R \Rightarrow (a,c) \in R $$

Example:

$$ R = \{(1,2),(2,3),(1,3)\} $$

Thus, R is transitive.

---

4. Anti-Symmetric Relation

A relation R is called anti-symmetric if

$$ (a,b) \in R \text{ and } (b,a) \in R \Rightarrow a = b $$

Example:

$$ R = \{(1,1),(2,2),(1,2)\} $$

Hence, R is anti-symmetric.

---

5. Equivalence Relation

A relation is called an equivalence relation if it satisfies the following three properties:

• Reflexive
• Symmetric
• Transitive

Example:

$$ R = \{(a,b) \mid a \equiv b \ (\text{mod } 2)\} $$

---

6. Partial Order Relation

A relation is a partial order relation if it is:

• Reflexive
• Anti-symmetric
• Transitive

Example:

The relation $\leq$ on the set of integers is a partial order relation.

---

Question 2

Give an example of a relation which is both symmetric and anti-symmetric.

Solution

Example

Let $A = \{1,2,3\}$. Define the relation

$$ R = \{(1,1),(2,2),(3,3)\} $$

Checking Symmetric Property

For every $(a,a) \in R$, $(a,a)$ also belongs to R. Hence, R is symmetric.

Checking Anti-Symmetric Property

Since only ordered pairs of the form $(a,a)$ are present, the condition of anti-symmetry is satisfied.

Final Conclusion

The relation

$$ R = \{(1,1),(2,2),(3,3)\} $$

is both symmetric and anti-symmetric.

Question 3

Define:
(i) Domain and Range of a relation
(ii) Void relation
(iii) Universal relation

Solution

(i) Domain and Range of a Relation

Let $R$ be a relation from set $A$ to set $B$.

The domain of $R$ is the set of all first elements of the ordered pairs in $R$.

$$ \text{Domain}(R) = \{ a \in A \mid (a,b) \in R \text{ for some } b \in B \} $$

The range of $R$ is the set of all second elements of the ordered pairs in $R$.

$$ \text{Range}(R) = \{ b \in B \mid (a,b) \in R \text{ for some } a \in A \} $$

Example:

Let $A = \{1,2,3\}$, $B = \{a,b\}$ and

$$ R = \{(1,a),(2,b)\} $$

Then,

• Domain$(R) = \{1,2\}$
• Range$(R) = \{a,b\}$

---

(ii) Void Relation

A relation is called a void relation if it contains no ordered pairs.

$$ R = \varnothing $$

Example:

If $A = \{1,2,3\}$, then

$$ R = \varnothing $$

is a void relation on set $A$.

---

(iii) Universal Relation

A relation is called a universal relation if it contains all possible ordered pairs from the Cartesian product $A \times B$.

$$ R = A \times B $$

Example:

Let $A = \{1,2\}$.

$$ R = \{(1,1),(1,2),(2,1),(2,2)\} $$

is a universal relation on $A$.

---

Question 4

Let $S = \{1,2,3,\dots,10\}$ and a relation $R$ on $S$ be defined as $R = \{(x,y) \mid x + y = 10\}$. Find the properties of the relation $R$.

Solution

Step 1: Write the Relation Explicitly

The ordered pairs satisfying $x + y = 10$ are:

$$ R = \{(1,9),(2,8),(3,7),(4,6),(5,5),(6,4),(7,3),(8,2),(9,1)\} $$

---

Step 2: Check Reflexive Property

A relation is reflexive if $(a,a) \in R$ for all $a \in S$.

Here, only $(5,5)$ belongs to $R$, but other elements like $(1,1)$, $(2,2)$ etc. are not present.

Therefore, R is not reflexive.

---

Step 3: Check Symmetric Property

A relation is symmetric if $(a,b) \in R$ implies $(b,a) \in R$.

Since $(1,9)$ and $(9,1)$ both belong to $R$, and this holds for all pairs,

R is symmetric.

---

Step 4: Check Transitive Property

For transitivity, if $(a,b)$ and $(b,c)$ belong to $R$, then $(a,c)$ must also belong to $R$.

However, $(1,9)$ and $(9,1)$ belong to $R$, but $(1,1)$ does not belong to $R$.

Hence, R is not transitive.

---

Step 5: Check Anti-Symmetric Property

A relation is anti-symmetric if $(a,b)$ and $(b,a)$ belonging to $R$ implies $a = b$.

Here, $(1,9)$ and $(9,1)$ belong to $R$ with $1 \neq 9$.

Therefore, R is not anti-symmetric.

---

Final Conclusion

• Reflexive: ❌
• Symmetric: ✅
• Transitive: ❌
• Anti-symmetric: ❌

---

---

Exam Tips

• The identity relation is always symmetric and anti-symmetric.
• Write definitions clearly before checking properties.
• Verify each property step by step in exams.

---