# Thread: equivalence relations and functions

1. ## 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

2. 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.

3. 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 $\displaystyle f(x) = 1 \;\forall x \in A$ would conform to f: A -> A.

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

$\displaystyle f(1) = \boxed{ }$

$\displaystyle f(2) = \boxed{ }$

...

$\displaystyle f(6) = \boxed{ }$

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.)

4. Originally Posted by undefined
I don't understand this statement.
$\displaystyle 1\to 1$
$\displaystyle 2\to 2$
$\displaystyle 3\to 3$
$\displaystyle 4\to 4$
$\displaystyle 5\to 5$
$\displaystyle 6\to 6$

This is what I meant. A mapped into A

5. Originally Posted by dwsmith
$\displaystyle 1\to 1$
$\displaystyle 2\to 2$
$\displaystyle 3\to 3$
$\displaystyle 4\to 4$
$\displaystyle 5\to 5$
$\displaystyle 6\to 6$

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.

6. 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.

7. 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.

8. 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 ($\displaystyle 1_A$ or $\displaystyle I_A$) 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,

$\displaystyle 1 \mapsto 2$
$\displaystyle 2 \mapsto 1$
$\displaystyle 3 \mapsto 3$
$\displaystyle 4 \mapsto 4$
$\displaystyle 5 \mapsto 5$
$\displaystyle 6 \mapsto 6$

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.