Let f: Z_5--->Z_5 be the function defined by f([a])=[2a+3]. Show that f is well-defined, and determine whether f is bijective.

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- Apr 17th 2008, 09:19 AMnatewalker205Bijection
Let f: Z_5--->Z_5 be the function defined by f([a])=[2a+3]. Show that f is well-defined, and determine whether f is bijective.

- Apr 17th 2008, 09:24 AMMathnasium
So your function looks like:

f(0) = 3

f(1) = 5 = 0

f(2) = 7 = 2

f(3) = 9 = 4

f(4) = 11 = 1.

In order for it to be well-defined, when we put each input, we should get exactly one output (i.e., f(a) can't be equal to two different things for a given a). Looks like our function is well-defined - when we put each thing in, we get out exactly one thing.

In order to be bijective, it has to be one-to-one and onto.

One-to-one means that each thing in the range has no more than one thing that maps to it. Clearly, this is true from the above table.

Onto means that everything in the range (Z_5) gets mapped to. Z_5 consists of 0, 1, 2, 3, and 4, and they all get mapped to by something.

So it looks both one-to-one and onto, and thus bijective.