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Introduction of Relations and Functions

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In this chapter, we are going to discuss what is a relation and how do we find the domain,   co-or domain and the range of the relation. Before we begin with what is a relation you need to have know what is a set, cartesian product, relation which would finally extended to a function. 
A set is a well defined collection of objects, that means whichever way you take you get a unique answer. 
Cartesian product is considered as `"A" xx "B"` which is defined to be all those ordered pairs where the first element comes from the first step and the second element comes from the second set and `xx` is referred as cross.
Example: A is a set having elements (1,2,3) and set B have elements (a,b,c,d).
`"A" xx "B"`= {(1,a),  (1,b), (1,c), (1,d), (2,a), (2,b), (2,c), (2,d), (3,a), (3,b), (3,c), (3,d)} this is our entire cartesian product which moves from A to B. Once we have the knowledge of this let's extend this to a relation. 
Relation is a connection between two sets A and B such that relation is a part or a subset of this cartesian product A cross B means here we have taken all possible ordered pairs from A to B whereas for a relation we'll be a little selective.
Example- Relation: all ordered pairs begining with an odd number
R= {(1,a),  (1,b), (1,c), (1,d), (3,a), (3,b), (3,c), (3,d)}, 
That means relation is a part of a cartesian product.
 R: A→B 
R⊆ A X B 
Example- A={1,2,3,4,5} B= {4,6,9}
R is a relation from A to B, R= {(x,y) difference between x and y is odd x∈A, y∈B} 
R= {(1,4), (1,6), (2,9), (3,4), (3,6), (5,4), (5,6)}
Domain- Set of all the 1st elements in the ordered pairs of R
Domain= {1,2,3,5}
Range- 2nd elements in the ordered pairs of R
Range= {4,6,9}
Co-Domain- is the 2nd set i.e B.

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