Please Wait... While Loading Full Video#### Class 11 Chapter 2 - Relations & Functions

### RELATIONS

`star` Relations

`color{blue} ul(mathtt ("Relation"))` `color{blue}:->` A relation `R` from a non-empty set `A` to a non-empty set `B` is a subset of the cartesian product `A × B.` The subset is derived by describing a relationship between the first element and the second element of the ordered pairs in `A × B.`

The second element is called `color{green} ("the image of the first element.")`

`color{purple} ✍️ \color{purple} \mathbf(KEY \ POINTS)`

`color{blue} ul(mathtt ("Domain"))color{blue}:->` The set of all first elements of the ordered pairs in a relation `R` from a set `A` to a set `B` is called `color{green} (" the domain of the relation R.") `

`color{blue} ul(mathtt ("Range")) color{blue}:->` The set of all second elements in a relation `R` from a set `A` to a set `B` is called `color{green} ("the range of the relation R.")`

`color{green} ✍️` The whole set `B` is called `color{green} (" the codomain of the relation R.")`

`color{green} ✍️` Note that `color{green} "range ⊆ codomain."`

● Consider the two sets `P = {a, b, c}` and `Q = {"Ali, Bhanu, Bi noy, Chandra, Divya"}`.

The cartesian product of `P` and `Q` has `15` ordered pairs which can be listed as

`P × Q = {(a, Ali), (a,Bhanu), (a, Bi noy), ..., (c, Divya)}.`

● We can now obtain a subset of `P × Q` by introducing a relation `R` between the first element `x` and the second element `y` of each ordered pair (x, y) as

`R= { (x,y): x \ \ "is the first letter of the name" \ \ y, x ∈ P, y ∈ Q}.`

Then `R = {(a, Ali), (b, Bhanu), (b, Bi noy), (c, "Chandra")}`

A visual representation of this relation `R` (called an arrow diagram) is shown in Fig 1.

`color{fuchsia} "Remarks"`

` (i)` A relation may be represented algebraically either by the Roster method or by the Set-builder method.

`(ii)` An arrow diagram is a visual representation of a relation.

`color{fuchsia} "Note : "` The total number of relations that can be defined from a set `A` to a set `B` is the number of possible subsets of `A × B.` If `color{fuchsia} [n(A ) = p]` and `color{fuchsia} [n(B) = q,]` then `color{fuchsia}[n (A × B) = pq]` and `color{fuchsia} "the total number of relations is"` `color{fuchsia} [2^(pq.)]`

The second element is called `color{green} ("the image of the first element.")`

`color{purple} ✍️ \color{purple} \mathbf(KEY \ POINTS)`

`color{blue} ul(mathtt ("Domain"))color{blue}:->` The set of all first elements of the ordered pairs in a relation `R` from a set `A` to a set `B` is called `color{green} (" the domain of the relation R.") `

`color{blue} ul(mathtt ("Range")) color{blue}:->` The set of all second elements in a relation `R` from a set `A` to a set `B` is called `color{green} ("the range of the relation R.")`

`color{green} ✍️` The whole set `B` is called `color{green} (" the codomain of the relation R.")`

`color{green} ✍️` Note that `color{green} "range ⊆ codomain."`

● Consider the two sets `P = {a, b, c}` and `Q = {"Ali, Bhanu, Bi noy, Chandra, Divya"}`.

The cartesian product of `P` and `Q` has `15` ordered pairs which can be listed as

`P × Q = {(a, Ali), (a,Bhanu), (a, Bi noy), ..., (c, Divya)}.`

● We can now obtain a subset of `P × Q` by introducing a relation `R` between the first element `x` and the second element `y` of each ordered pair (x, y) as

`R= { (x,y): x \ \ "is the first letter of the name" \ \ y, x ∈ P, y ∈ Q}.`

Then `R = {(a, Ali), (b, Bhanu), (b, Bi noy), (c, "Chandra")}`

A visual representation of this relation `R` (called an arrow diagram) is shown in Fig 1.

`color{fuchsia} "Remarks"`

` (i)` A relation may be represented algebraically either by the Roster method or by the Set-builder method.

`(ii)` An arrow diagram is a visual representation of a relation.

`color{fuchsia} "Note : "` The total number of relations that can be defined from a set `A` to a set `B` is the number of possible subsets of `A × B.` If `color{fuchsia} [n(A ) = p]` and `color{fuchsia} [n(B) = q,]` then `color{fuchsia}[n (A × B) = pq]` and `color{fuchsia} "the total number of relations is"` `color{fuchsia} [2^(pq.)]`

Q 3087880787

Let `A = {1, 2, 3, 4, 5, 6}`. Define a relation R from A to A by `R = {(x, y) : y = x + 1 }`

(i) Depict this relation using an arrow diagram.

(ii) Write down the domain, codomain and range of R.

(i) Depict this relation using an arrow diagram.

(ii) Write down the domain, codomain and range of R.

(i) By the definition of the relation, `R = {(1,2), (2,3), (3,4), (4,5), (5,6)}.`

The corresponding arrow diagram is shown in Fig

(ii) We can see that the

domain `={1, 2, 3, 4, 5,}` Similarly, the range `= {2, 3, 4, 5, 6}` and the codomain `= {1, 2, 3, 4, 5, 6}.`

Q 3067080885

The Fig shows a relation between the sets P and Q. Write this relation (i) in set-builder form, (ii) in roster form. What is its domain and range?

It is obvious that the relation R is “x is the square of y”.

(i) In set-builder form, `R = {(x, y) : x` is the square of `y, x ∈ P, y ∈ Q}`

(ii) In roster form, `R = {(9, 3), (9, –3), (4, 2), (4, –2), (25, 5), (25, –5)}`

The domain of this relation is `{4, 9, 25}.` The range of this relation is `{– 2, 2, –3, 3, –5, 5}.`

Note that the element 1 is not related to any element in set P. The set Q is the codomain of this relation.

Q 3007191088

Let `A = {1, 2}` and `B = {3, 4}.` Find the number of relations from A to B.

We have,

`A × B = {(1, 3), (1, 4), (2, 3), (2, 4)}.` Since `n (A×B ) = 4,` the number of subsets of `A×B` is `2^4.` Therefore, the number of

relations from A into B will be `2^4`.

Q 3067291185

Let `N` be the set of natural numbers and the relation `R` be defined on `N` such that `R = {(x, y) : y = 2x, x, y ∈ N}.`

What is the domain, codomain and range of `R`? Is this relation a function?

What is the domain, codomain and range of `R`? Is this relation a function?

The domain of R is the set of natural numbers `N.` The codomain is also `N`. The range is the set of even natural numbers.

Since every natural number n has one and only one image, this relation is a function.