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injective or surjective whats the difference?

Mathematics
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(a) What does it mean for a function to be injective (1-1)? (b) What does it mean for a function to be surjective (onto)? (c) Give an example of a function on the integers that is 1-1 but not onto. (d) Give an example of a function on the integers that is onto but not 1-1.

Apr 21st, 2015

(a) A function f: X → Y is said to be injective (1-1) if f(x) = f(y) implies x = y for all x,y ϵ X,

or equivalently, x ≠ y implies f(x) ≠ f(y) for all x,y ϵ X.

(b) A function f: X → Y is said to be surjective (onto) if for all y ϵ Y, there exists x ϵ X such that f(x) = y.

(c) An example of a function f: Z → Z that is 1-1 but not onto is:

f(x) = 2x

It is easy to check that f is injective (1-1) since f(x) = f(y) implies 2x = 2y implies x = y. However, f is not surjective (onto) since if y is an odd integer, then there is no such x satisfying f(x) = y.

(d) We can construct a function f: Z → Z that is onto but not 1-1 as follow:

f(x) = x for x < 0 (for example f(-1) = -1, f(-2) = -2, etc)

f(0) = 0

f(1) = 0

f(x) = x - 1 for x > 1 (for example, f(2) = 1, f(3) = 2, f(4) = 3, etc)

It is obvious to check that for any y ϵ Z , we can always find x ϵ Z such that f(x) = y, so f is onto. However, f is not 1-1 because we have f(0) = f(1) but 0 ≠ 1.

*Z is denoted to be the set of integers. In question (c) and (d), we have that the domain X and the range Y of the function f are equal to Z.


Apr 21st, 2015

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