Mathematics RELATIONS
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### 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.

Solution:

(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?

Solution:

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.

Solution:

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?

Solution:

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.