Define and give examples of injective surjective and bijective functions. Check the injectivity and surjectivity of the following function f: NN given by f(x)=x2
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injective function Definition:
A function f: A → B is said to be a one - one function or injective mapping if different elements of A have different f images in B. A function f is injective if and only if whenever f(x) = f(y), x = y. Example: f(x) = x + 9 from the set of real number R to R is an injective function. When x = 3,then :f(x) = 12,when f(y) = 8,the value of y can only be 3,so x = y.
(ii) surjective function Definition: If the function f:A→B is such that each element in B (co - domain) is the ‘f’ image of at least one element in A , then we say that f is a function of A ‘onto’ B .Thus f: A→B is surjective if, for all b ∈ B, there are some a ∈ A such that f(a) = b.
Example: The function f(x) = 2x from the set of natural numbers N to the set of non negative even numbers is a surjective function.
(iii) bijective function Definition: A function f (from set A to B) is bijective if, for every y in B, there is exactly one x in A such that f(x) = y. Alternatively, f is bijective if it is a one - to - one correspondence between those sets, in other words, both injective and surjective.
Example: If f(x) = x2,from the set of positive real numbers to positive real numbers is both injective and surjective. Thus, it is a bijective function.
hence f is injective, for some elements like, 2,3 etc has no preimage in N such that f(x)=2 hence not surjective.