# Relations

**Relation: Meaning**

Let *A* and *B* be two sets, then a relation R from A to B is a subset of *A ×B*.

Thus, *R* is a relation from A to B . ⇔ R ⊆ A × B

Recall that *A × B *is a set of all ordered pairs whose first member is from the set *A *and second member is from *B *i.e.,

A × B = {(x,y) : x ∈ A and y ∈ B }

If *R* is a relation from a non-empty set *A* to a non-empty set *B *and if (*a, b*) ∈ *R, *then we write *aRb* which is read as *a *is related to *b *by the relation *R. *If (*a, b*) ∉ *R* then we say that *a* is not related to *b* by the relation *R*. In other words, a relation is a set of inputs and outputs, often written as ordered pairs (input, output). We can also represent a relation as a mapping or a graph. For example, a relation can be represented as:

**Mapping diagram of Relation**

**Domain and Range of relation:** Let *R* be a relation from a set *A* to set *B.* Then the set *A *is called the domain of *R*. The set of all second elements of the ordered pairs in *R* is called the range of *R*. The set *B *is called the co-domain of the relation *R.*

# Types of Relations

**(i)** **Universal Relation :** If *A* be any set, then R = A × A ⊆ A × A and so it is a relation on *A*. This relation is called the universal relation on *A*.

**(ii) Empty (or Void) Relation :** If *A* be any set, then Φ ⊆ A × A and so it is a relation on *A*. This relation is called Empty (or void) relation.

**(iii) Reflexive Relation :** A relation *R* is said to be reflexive, if every element of *A* is related to itself.

Thus, *R* is reflexive ⇔ (a,a) ∈ *R, *for all a ∈ A .

**(iv) ****Symmetric Relation :** A relation *R* on a set *A* is said to be a symmetric if (a,b) ∈ R ⇔ (b,a) ∈ R for all a,b ∈ A

**(v) Transitive Relation :** A relation *R *on any set *A* is said to be transitive if (a,b) ∈ R, (b,c)∈ R ⇒(a,c)∈ R,for all a,b,c ∈ A.

i.e., aRb and bRc ⇒ aRC for all a,b,c ∈ aRC

**Equivalence Relation :** A relation *R* on a set *A* is said to be an equivalence relation on* A* if

(a) it is reflexive

(b) it is symmetric

(c) it is transitive

# Functions

**Function : Meaning**

Let *A* and *B* be two non-empty sets, then a rule *f* which associates each element of *A* with a unique element of *B* is called a function or mapping from *A* to *B*.

If *f* is a mapping from *A *to *B, *we write f : A → B (read as *f* is a mapping from *A *to *B*).

If *f* associates x ∈ A to y ∈ B then we say that *y* is the image of the element *x *under the map (or function) *f* and we write *y *= *f *(*x*). And, the element *x* is called the pre-image of *y*.

### Types of Functions

**(1) One-One Function (Injective) :** A function* *f : A → B* *is said to be a one-one function or an injective if different elements of *A* have different images in *B*.

Thus, f : A → B is one-one.

⇔ a≠b ⇒ f(a) ≠ f(b) for all a,b ∈ A or f(a) = f(b)⇔ a=b for all a,b ∈ A

**(2) Onto-Function (Surjective) :** A function f : A → B is said to be an onto function or surjective if and only if each element of *B* is the image of some element of *A* i.e. for every element y ∈ B there exists some x ∈ A such that *y *= *f *(*x*). Thus *f* is onto if range of

*f *= co-domain of *f*.

**(3) One-one and onto Function (Bijective Function) :** A function is a bijective if it is one-one as well as onto

# Composition of Functions

Suppose *A, B* and *C* be three non-empty sets and let f : A → B, g : B → C be two functions. since *f* is a function from *A* to *B*, therefore for each x ∈ A there exists a unique element

f(x) ∈ B Again, since *g* is a function from *B* to *C*, therefore corresponding to f(x) ∈ B there exists a unique element g(f(x)) ∈ C Thus, for each fx ∈ A there exists a unique element g(f(x)) ∈ C .

From above discussion it follows that when *f* and *g* are considered together they define a new function from *A* to *C*. This function is called the composition of *f* and *g* and is denoted by *gof*.

i.e. *gof*(*x*) = *g*(*f *(*x*)), for all x ∈ A

# Inverse of a function

If *f *and *g* be two functions satisfying

*f *(g(*x*)) = *x, *for every *x* in the domain of *g*, and *g*(*f*(*x*)) = *x* for every *x *in the domain of *f*.

We say that *f* is the inverse of *g* and also, *g* is the inverse of *f*. We write f^{ = }g^{−1}or g = f^{−1}

**Tips:** To find the inverse of *f*, write down the equation *y*= *f*(*x*) and then solve *x* as a function of *y*.

Thus the resulting equation is x = f^{−1}(y)^{ .}

**Note :** (i) If f is one-one function then *f* has inverse defined on its range

(ii) If* f* and *g* are one-one function then (fog)^{−1 }=^{ }g^{−1}of^{−1}

(iii) If f is bijective then f^{−1} is bijective and (f^{−1})^{−1 }= f.

## Binary Operation and Properties of Binary Operations

**Binary Operation :** A binary operation ∗ on a set *A *is a function ∗: A × A . We denote ∗ (*a, b*) by *a *∗ *b.*

It follows from the definition of a function that a binary operation on a set *A *associates each ordered pair (*a, b*) ,

(a,b) ∈ A × A to a unique element (*a, b*) in *A.*

**Properties of Binary Operations**

**(i) ****Commutativity:** A binary operation ∗ on a set of *A *is said to be commutative binary operation, if

a ∗ b =b ∗ a for all a,b ∈ A

The binary operations addition (+) and multiplication (×) are commutative binary operation on *z.*

**(ii) ****Associativity:** A binary operation ∗: A × A → Ais said to be associative if

(a∗b)∗c =a∗(b∗c) , ∀ a,b,c ∈ A

The binary operation of addition (+) and multiplication (×) are associative binary operation on *z. *However, the binary operation of subtraction is not associative binary operation.

**(iii) ****Identity Element:** Let ′∗′ be a binary operation on a set *A *and if there exists an element e ∈ A, such that a∗e = a = e∗a, for all a ∈ A, then *e *is called an identity element for the binary operation ′∗′ on set *A *and it is unique.

**(iv) ****Inverse of an element:** Let ′∗′ be a binary operation on a set *A, *and let *e *be the identity element in *A *for the binary operation ∗ on *S.*Then, an element a ∈ A is called an invertible element if there exists an element b ∈ A such that a×b = e = b∗a.

The element *b *is called the inverse of an element *a, *and it is unique.

**Note:** The inverse of *a* element is generally denoted by a^{-1}

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