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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. [inline]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 [/inline] 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 [inline]ceil[/inline] rounds a real number up to the nearest integer, so [inline]ceil(5)=5[/inline], [inline]ceil(5.1)=6[/inline], 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:  
Thank you for replying. 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: [inline] \displaystyle f : \mathbb{N} \rightarrow \mathbb{N} ; \space f : x \mapsto \left\lceil\frac{x}{2}\right\rceil \forall x \in \mathbb{N}[/inline] For the second function I'm guessing I'll have to set the condition that when x = 1 [inline]f : x \mapsto x[/inline], and all the other natural numbers > 1 are [inline] f : x \mapsto x  1[/inline]? For this can I just set the two conditions like I did in my first post with the semicolons, 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: [inline]f: N \rightarrow N[/inline] [inline]x \mapsto x/2[/inline] if [inline]x[/inline] is even [inline]x \mapsto (x+1)/2[/inline] if [inline]x[/inline] 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. Thanks for your help 
