Thread: equivalence relations and functions

im having trouble on a problem

A={1,2,3,4,5,6} f:A->A

i need to find an example where f is bijective, but not 1A

Originally Posted by kdigga

im having trouble on a problem

A={1,2,3,4,5,6} f:A->A

i need to find an example where f is bijective, but not 1A

If you map A into A, then f is both one to one and onto.

Originally Posted by dwsmith

If you map A into A, then f is both one to one and onto.

I don't understand this statement. f: A -> A doesn't require that A be the image but only the codomain, so for example the function would conform to f: A -> A.

But I also don't see the difficulty in finding a bijective function. Just write down





...



and fill in with values from A such that you use every single element, then it will be bijective. (Sorry the rectangles are a bit uncentered vertically, couldn't find the proper LaTeX.)

Originally Posted by undefined

I don't understand this statement.

This is what I meant. A mapped into A

Originally Posted by dwsmith

This is what I meant. A mapped into A

Oh. I thought the original poster was referring to that as 1A and was asking for a different one.

Originally Posted by undefined

Oh. I thought the original poster was referring to that as 1A and was asking for a different one.

I have no idea what 1A is supposed to mean.

Originally Posted by dwsmith

I have no idea what 1A is supposed to mean.

Yeah, it was just a guess based on context. Well in any case it's the identity function. I'd never heard "f maps A into A" meaning f is the identity function, and seemed f: A -> A would naturally be read as "f maps A into A" hence my original response.

Originally Posted by undefined

Yeah, it was just a guess based on context. Well in any case it's the identity function. I'd never heard "f maps A into A" meaning f is the identity function, and seemed f: A -> A would naturally be read as "f maps A into A" hence my original response.

The word `into' is sometimes used to mean `injection' (much like `onto'). When there exists an injection, f, from A to B then it means you can find a copy of A in B, and so f maps A into B.

I would also presume, as did undefined, that 1A means the identity function ( or ) as this would be too easy an answer and noone would exclude any other bijection.

Now, bijections between finite sets are just permutations. So, permute your elements. For example,








will be a function which switches 1 and 2. It is thus a permutation, as it permutes these two elements and keeps everything else fixed.