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EJChoi
Poster

Post Number: 6
 Posted on Saturday, 29 June, 2013 - 11:57 am:

I've got an exercise which asks me to define a function from the natural numbers to themselves which is surjective but not injective.

I've thought of a function which pairs 1,2 with 1 and 3,4 with 2, and 5,6 with 3 etc. But how could I represent this symbolically?

I did something like this.

$f: \mathbb{N} \rightarrow \mathbb{N} ; \space f: x \mapsto \frac{x}{2} \space | \space x,y \in \mathbb{N}, \space \frac{x}{2} = y \space; f: x \mapsto \frac{x + 1}{2} \space | \space x,y \in \mathbb{N} \space \frac{x + 1}{2} = y$

Are there useful and relevant quantifiers that would be of use in these situations?
Quixote (Tim King)
Frequent poster

Post Number: 192
 Posted on Saturday, 29 June, 2013 - 01:55 pm:

Hint: The function $ceil$ rounds a real number up to the nearest integer, so $ceil(5)=5$, $ceil(5.1)=6$, etc.

Can you use this to express your function succinctly?

A different function which does the trick is done as follows:
Map both 1 and 2 to 1 as you suggest
map 3 to 2,
map 4 to 3, etc.
See if you can write that out symbolically.
EJChoi
Poster

Post Number: 7
 Posted on Saturday, 29 June, 2013 - 04:50 pm:

I've only used the floor function before in a problem. I didn't know it was a conventional notation.

If we set it up with the ceiling function, then:

$\displaystyle f : \mathbb{N} \rightarrow \mathbb{N} ; \space f : x \mapsto \left\lceil\frac{x}{2}\right\rceil \forall x \in \mathbb{N}$

For the second function I'm guessing I'll have to set the condition that when x = 1 $f : x \mapsto x$, and all the other natural numbers > 1 are $f : x \mapsto x - 1$? For this can I just set the two conditions like I did in my first post with the semi-colons, or will I have to employ the squiggly bracket?
Quixote (Tim King)
Frequent poster

Post Number: 193
 Posted on Saturday, 29 June, 2013 - 06:22 pm:

While the ceiling function is fairly standard, you would probably want to define it if you used it (say) in an article.

I don't think that there is anything wrong with the curly bracket (aside from the fact that it's hard to LaTeX). All that matters is that you decribe your function in a completely nonambiguous manner. For example, you could have said, for your original function:

$f: N \rightarrow N$
$x \mapsto x/2$ if $x$ is even
$x \mapsto (x+1)/2$ if $x$ is odd

This is perfectly nonambiguous, and much clearer than a huge load of quantifiers. Unless you are doing a lecture course in formal logic, I would prefer what I have said above to a long collection of quantifiers with no written English.

EJChoi
Poster

Post Number: 8
 Posted on Saturday, 29 June, 2013 - 06:26 pm:

Oh ok. I guess I got overexcited with the quantifiers.