Mathematics RELATIONS
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`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.)]`
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) 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?


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.