example of a function f:z-->z that is surjective but not injective
To find an example of a function that is surjective but not injective, we need to find a function that maps from the integers (Z) to the integers (Z) and satisfies these conditions:
1. Surjective: Every integer in the codomain (Z) has a corresponding integer in the domain (Z) that maps to it.
2. Not injective: There are two or more integers in the domain (Z) that map to the same integer in the codomain (Z).
One such example is the function f(x) = x^2, where x is an integer. Let's verify that this function is surjective but not injective.
To check surjectivity:
For any given integer n in the codomain (Z), we need to find an integer m in the domain (Z) such that f(m) = n.
Since squaring any integer results in a non-negative integer (including zero), we can see that for every non-negative integer n, we can find an integer m such that m^2 = n. Therefore, this function is surjective for the non-negative integers.
To check injectivity:
A function is injective if each element in the domain maps to a distinct element in the codomain. In this case, since x^2 is not a one-to-one function, it is not injective.
For example, f(2) = 4 and f(-2) = 4, so two different elements in the domain (2 and -2) map to the same element in the codomain (4).
Therefore, the function f(x) = x^2 is an example of a function that is surjective but not injective when mapping from the integers (Z) to the integers (Z).