functions and types of functions

UNIT - FUNCTIONS

FUNCTION :- Let ‘A’ and ‘B’ be the two non-empty sets. A function ‘f’ from ‘A’ to ‘B’ is denoted by
f:A→B is a rule which assigns to every element of ‘A’ ,a unique element of ‘B’.

#) ‘A’ is called domain of function ‘f’.

#) ‘B’ is called codomain of function ‘f’.

#) f(A) :={f(x) ∈B | x∈ A } is called range of function ‘f’.

#) f(x) =y , than y is called image of x under f and x is called a pre image of y under f.

#)Range ⊆ Codomain , |Range |≤|Domain|, |Range |≤ Minimum{|Domain|,|Codomain| }

1) Is f a function from ‘A’ to ‘B’?

Explanation:- If ∀ x,y ∈ A , a) f(x) and f(y) exist in B( i.e. existence of image for every x ∈ A) and

b) x=y ⇒ f(x) = f(y) (i.e. uniqueness of image or in simple words any single
element in A cannot have more than one image in B).

than , f:A→B is a function.



FUNCTIONS




ONTO MANY-ONE INTO
ONE-ONE
FUNCTION FUNCTION FUNCTION
FUNCTION
(INJECTIVE (SURJECTIVE
FUNCTION) FUNCTION)




I) One-one function :- ∀ x,y ∈ A , if f(x) = f(y) ⇒ x=y .
In one – one function the cardinality of range is equal to domain i.e |RANGE|=|
DOMAIN| and the cardinality of domain is less than equal to codomain i.e |
DOMAIN |≤|CODOMAIN|. Hence, there cannot be any one-one function from
uncountable set to a countable set.