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RELATION AND
FUNCTION
CLASS - 12
MATHEMATICS
Relation -
If A and B are two non-empty sets, then a relation R from A to B is a
subset of A x B. If R ⊆ A x B and (a, b) ∈ R, then we say that a is
related to b by the relation R, written as aRb.
Domain and Range of a Relation-
Let R be a relation from a set A to set B. Then, 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 belonging to R is called the range
of R.
Thus, domain of R = {a : (a , b) ∈ R} and range of R = {b : (a, b) ∈ R}
Types of Relations-
(i) Void Relation
As Φ ⊂ A x A, for any set A, so Φ is a relation
on A, called the empty or void relation.
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(ii) Universal Relation
Since, A x A ⊆ A x A, so A x A is a relation on A,
called the universal relation.
(iii) Identity Relation The relation IA = {(a, a) : a ∈ A} is called
the identity relation on A.
(iv) Reflexive Relation A relation R is said to be reflexive
relation, if every element of A is related to itself. Thus, (a, a) ∈ R, ∀ a
∈ A = R is reflexive.
(v) Symmetric Relation
A relation R is said to be symmetric relation, iff (a, b) ∈ R (b, a) ∈
R,∀ a, b ∈ A i.e., a R b ⇒ b R a,∀ a, b ∈ A ⇒ R is symmetric.
(vi) Anti-Symmetric Relation A relation R is said to be
anti-symmetric relation, iff (a, b) ∈ R and (b, a) ∈ R ⇒ a = b,∀ a, b
∈ A.
(vii) Transitive Relation A relation R is said to be transitive
relation, iff (a, b) ∈ R and (b, c) ∈ R ⇒ (a, c) ∈ R, ∀ a, b, c ∈ A
(viii) Equivalence Relation A relation R is said to be an
equivalence relation, if it is simultaneously reflexive, symmetric and
transitive on A.
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RELATION AND
FUNCTION
CLASS - 12
MATHEMATICS
Relation -
If A and B are two non-empty sets, then a relation R from A to B is a
subset of A x B. If R ⊆ A x B and (a, b) ∈ R, then we say that a is
related to b by the relation R, written as aRb.
Domain and Range of a Relation-
Let R be a relation from a set A to set B. Then, 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 belonging to R is called the range
of R.
Thus, domain of R = {a : (a , b) ∈ R} and range of R = {b : (a, b) ∈ R}
Types of Relations-
(i) Void Relation
As Φ ⊂ A x A, for any set A, so Φ is a relation
on A, called the empty or void relation.
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(ii) Universal Relation
Since, A x A ⊆ A x A, so A x A is a relation on A,
called the universal relation.
(iii) Identity Relation The relation IA = {(a, a) : a ∈ A} is called
the identity relation on A.
(iv) Reflexive Relation A relation R is said to be reflexive
relation, if every element of A is related to itself. Thus, (a, a) ∈ R, ∀ a
∈ A = R is reflexive.
(v) Symmetric Relation
A relation R is said to be symmetric relation, iff (a, b) ∈ R (b, a) ∈
R,∀ a, b ∈ A i.e., a R b ⇒ b R a,∀ a, b ∈ A ⇒ R is symmetric.
(vi) Anti-Symmetric Relation A relation R is said to be
anti-symmetric relation, iff (a, b) ∈ R and (b, a) ∈ R ⇒ a = b,∀ a, b
∈ A.
(vii) Transitive Relation A relation R is said to be transitive
relation, iff (a, b) ∈ R and (b, c) ∈ R ⇒ (a, c) ∈ R, ∀ a, b, c ∈ A
(viii) Equivalence Relation A relation R is said to be an
equivalence relation, if it is simultaneously reflexive, symmetric and
transitive on A.
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