# Different Types of Relations

###
Void Relation

An empty relation (or void relation) is one in which there is no
relation between any elements of a set. For example, if set A = {1, 2, 3} then,
one of the void relations can be R = {x, y} where, |x – y| = 8. For empty
relation,

R = φ ⊂ A × A

Universal Relation

A universal (or full relation) is a type of relation in which every
element of a set is related to each other. Consider set A = {a, b, c}. Now one
of the universal relations will be R = {x, y} where, |x – y| ≥ 0. For universal
relation,

R = A × A

Identity Relation

In an identity relation, every element of a set is related to itself
only. For example, in a set A = {a, b, c}, the identity relation will be I =
{a, a}, {b, b}, {c, c}. For identity relation,

I = {(a, a), a ∈ A}

Inverse Relation

Inverse relation is seen when a set has elements which are inverse pairs
of another set. For example if set A = {(a, b), (c, d)}, then inverse relation
will be R^{-1} = {(b, a), (d, c)}. So, for an inverse relation,

R^{-1} = {(b, a): (a, b) ∈ R}

Reflexive Relation

In a reflexive relation, every element maps to itself. For example,
consider a set A = {1, 2,}. Now an example of reflexive relation will be R =
{(1, 1), (2, 2), (1, 2), (2, 1)}. The reflexive relation is given by-

(a, a) ∈ R

Symmetric Relation

In a symmetric relation, if a=b is true then b=a is also true. In other
words, a relation R is symmetric only if (b, a) ∈ R is true when (a,b) ∈ R. An example of symmetric relation will be R = {(1, 2), (2, 1)} for a
set A = {1, 2}. So, for a symmetric relation,

aRb ⇒ bRa, ∀ a, b ∈ A

Transitive Relation

For transitive relation, if (x, y) ∈ R, (y, z) ∈ R, then (x, z) ∈ R. For a transitive relation,

aRb and bRc ⇒ aRc ∀ a, b, c ∈ A

Equivalence Relation

If a relation is reflexive, symmetric and transitive at the same time it
is known as an equivalence relation.

Sample Problem 1

The relation R defined in A = {1, 2,3} by aRb, if |$a^2 + b^2$| $\leq$ 5. Which of the following is false?

(a) R = {(1, 1), (1, 2), (2, 1), (2, 2), (2, 3), (3, 2), (3,3)}

(b) $R^{-1}=R$

(c) Domain of R = {1, 2, 3}

(d) Range of R = {5}

Answer:

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

$R^{-1}$ = {(y,x):(x,y) $\in$ R}

$R^{-1}$ = {(1, 1), (1, 2), (2, 1), (2, 2), (2, 3), (3, 2), (3,3)} = R

Domain of R = {x: (x, y) $\in$ R} = {1, 2, 3} and range of R = {y: (x, y) $\in$ R} = {1, 2, 3}

Sample Problem 2

Let R be the relation from A = {2, 3, 4, 5} to
B = {3 ,6 , 7, 10} defined by ‘x divides y’, then $R^{-1}$ is
equal to

(a) {(6, 2), (3, 3)}

(b) {(6, 2), (10, 2), (3, 3), (6, 3),(10, 5)}

(c) {(6, 2), (10, 2),(3, 3), (6, 3)}

(d) None of the above

Answer: (b)

Given:

A = {2, 3, 4, 5} to B = {3 ,6 , 7, 10}

So, R = {(2, 6), (2, 10), (3, 3), (3, 6),(5, 10)}

⇒ $R^{-1}$ = {(6, 2), (10, 2), (3, 3), (6, 3),(10, 5)}

**Contents:**

Void Relation

An empty relation (or void relation) is one in which there is no relation between any elements of a set. For example, if set A = {1, 2, 3} then, one of the void relations can be R = {x, y} where, |x – y| = 8. For empty relation,

R = φ ⊂ A × A

Universal Relation

A universal (or full relation) is a type of relation in which every element of a set is related to each other. Consider set A = {a, b, c}. Now one of the universal relations will be R = {x, y} where, |x – y| ≥ 0. For universal relation,

R = A × A

Identity Relation

In an identity relation, every element of a set is related to itself only. For example, in a set A = {a, b, c}, the identity relation will be I = {a, a}, {b, b}, {c, c}. For identity relation,

I = {(a, a), a ∈ A}

Inverse Relation

Inverse relation is seen when a set has elements which are inverse pairs
of another set. For example if set A = {(a, b), (c, d)}, then inverse relation
will be R^{-1} = {(b, a), (d, c)}. So, for an inverse relation,

R^{-1} = {(b, a): (a, b) ∈ R}

Reflexive Relation

In a reflexive relation, every element maps to itself. For example, consider a set A = {1, 2,}. Now an example of reflexive relation will be R = {(1, 1), (2, 2), (1, 2), (2, 1)}. The reflexive relation is given by-

(a, a) ∈ R

Symmetric Relation

In a symmetric relation, if a=b is true then b=a is also true. In other words, a relation R is symmetric only if (b, a) ∈ R is true when (a,b) ∈ R. An example of symmetric relation will be R = {(1, 2), (2, 1)} for a set A = {1, 2}. So, for a symmetric relation,

aRb ⇒ bRa, ∀ a, b ∈ A

Transitive Relation

For transitive relation, if (x, y) ∈ R, (y, z) ∈ R, then (x, z) ∈ R. For a transitive relation,

aRb and bRc ⇒ aRc ∀ a, b, c ∈ A

Equivalence Relation

If a relation is reflexive, symmetric and transitive at the same time it is known as an equivalence relation.

Sample Problem 1

The relation R defined in A = {1, 2,3} by aRb, if |$a^2 + b^2$| $\leq$ 5. Which of the following is false?

(a) R = {(1, 1), (1, 2), (2, 1), (2, 2), (2, 3), (3, 2), (3,3)}

(b) $R^{-1}=R$

(c) Domain of R = {1, 2, 3}

(d) Range of R = {5}

Answer:

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

$R^{-1}$ = {(y,x):(x,y) $\in$ R}

$R^{-1}$ = {(1, 1), (1, 2), (2, 1), (2, 2), (2, 3), (3, 2), (3,3)} = R

Domain of R = {x: (x, y) $\in$ R} = {1, 2, 3} and range of R = {y: (x, y) $\in$ R} = {1, 2, 3}

Sample Problem 2

Let R be the relation from A = {2, 3, 4, 5} to
B = {3 ,6 , 7, 10} defined by ‘x divides y’, then $R^{-1}$ is
equal to

(a) {(6, 2), (3, 3)}

(b) {(6, 2), (10, 2), (3, 3), (6, 3),(10, 5)}

(c) {(6, 2), (10, 2),(3, 3), (6, 3)}

(d) None of the above

Answer: (b)

Given:

A = {2, 3, 4, 5} to B = {3 ,6 , 7, 10}

So, R = {(2, 6), (2, 10), (3, 3), (3, 6),(5, 10)}

⇒ $R^{-1}$ = {(6, 2), (10, 2), (3, 3), (6, 3),(10, 5)}

**Contents:**

## Post a Comment for "Different Types of Relations"