Relations can be defined as set of ordered pairs of input values with output values, such that ‘P’ and ‘Q’ be two sets, a binary relation from P to Q is a subset of P´Q.
It is a rule that produces one or more output numbers for every valid input number.
The set of input values i.e. first element in a relation is called the domain. These are also called the independent variable.
Domain = 1, 2, 3, 4
These are ‘x’ values written in a set from smallest to largest.
The set of output values i.e. the second element in a relation is called the range. These are also called the dependent variables.
Range = 2, 4, 6, 8
There are many ways to represent relations like:
These are ‘y’ values written in a set from smallest to largest.
This relation can be written (1,6), (2,2), (3,4), (4,8).
Types of Relations:
1. Identity Relations => A relation ‘R’ on a set ‘P’ is called identity if (p, p): a ÎP.
2. Reflexive Relation => A relation ‘R’ on a set ‘P’ is called reflexive if (p, p) ÎR for every element pÎP.
3. Symmetric Relation => A relation ‘R’ on a set ‘P’ is called symmetric if (q, p) ÎR whenever (p, q)ÎR for all p, qÎP.
4. Anti symmetric Relation => A relation ‘R’ on a set ‘P’ is called anti symmetric if
p = q whenever (a, b) ÎR and (b, a) ÎR.
5. Transitive Relation => A relation ‘R’ on a set ‘P’ is called transitive if (p, q) ÎR and (q, c) ÎR, then (p, c) ÎR for p, q, cÎP.
Now we will discuss functions:
A function is a well-behaved relation between a set of inputs and a set of potential outputs with the property that each input is related to exactly one output.
It always gives a single output number for every valid input number
Argument -> the input to a function.
Value -> the output.
Note that in relations and functions, a function is always a relation but it is not necessary that a relation is always a function.
Types of Functions which are included in relation and function:
1. Injective Function => Let ‘f’ be a function from P to Q then ‘f’ is called injection if and only if different elements of ‘P’ have different images in ‘Q’.
2. Surjective function => Let ‘f’ be a function from P to Q then ‘f’ is called surjection if and only if each elements of ‘Q’ is the image of at least one element of ‘P’, i.e., if and only if range of ‘f’ is equals to ‘Q’.
3. Bijective Function => Let ‘f’ be a function from P to Q then ‘f’ is called bijection if and only if ‘f’ is both one-one and onto.
This is all about relation and function in mathematics.