# Relations and Functions Solutions and Shortcuts

Relations and Functions Examples - Page 2
Relations and Functions Questions - Page 4
Relations and Functions Video Lecture - Page 5

Consider the two sets P = {a,b,c} and Q = {Ali, Bhanu, Binoy, 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, Binoy), ...,
(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 ${Y,X\in P,Y\in Q}$ .Then R = {(a, Ali), (b, Bhanu), (b, Binoy), (c, Chandra)}A visual representation of this relation R (called an arrow diagram) is shown in Fig 2.4

Definition: 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 the image of the first element. or
The set of all first elements of the ordered pairs in a relation R from a set A to a set B is called the domain
of the relation R.. or
Definition: The set of all second elements in a relation R from a set A to a set B is called the range of the relation R. The whole set B is called the codomain of the relation R. Note that range ⊆ co-domain.

Note:

• A relation may be represented algebraically either by the Roster method or by the Set-builder method.
• An arrow diagram is a visual representation of a relation.

Important Point: The total number of relations can be define from a set A to a set B is the number of possible subset of a A*B. If n(A) = P and n (B) = q , then n(A*B) = pq and the total number of relation is 2pq

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