Ken writes:

My dad, who is a mathematician (or rather a statistician), occasionally reads this blog so partly for his amusement I thought I’d share the following vaguely mathematical thought I’ve had lately.

I heard on the radio about a scheme devised to cheer school children up in January to help them through the post Christmas lull. Every child would anonymously write a note to another child complimenting them on all the positive things about them. Phew (sucks air through teeth), I thought, as I heard this. This will be a tricky thing to manage. In mathematical terms, assuming each child only makes one card, that means you have a 1 to 1 function, but what you really need here is an ‘onto’ function. It might cheer people up to *make* positive cards, but it will matter more to *receive* one. Or rather, it would make children unhappy if they didn’t get one. If you let children pick who they will give the cards to, some children might get two or more cards and some none at all. An each child gives a card function is a 1 to 1 function. An each child gets a card function is an onto function. A bijection is a function that is both, which is what the poor teachers will have to ensure applies: that each child gives a card and each child gets one.

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The function ought also to be irreflexive, I’d have thought.

You are quite right Chris!

“A sends a card to B” is a relation R on the set of children but it seems to lack any general properties.

* A may send cards to more than one child so R need not be a function.

* B may receive cards from more than one child, so R need not be 1-1.

* As Ken notes, R need not be onto.

*

PaceChris, R need not be irreflexive as a lonely but smart child may send themself an anonymous card, thus avoiding the stigma of not getting a card.Shows how much I know: I thought 1 to 1 ness needed no more than that every A gives a card to some B (not some unique B). I guess, thinking about it, that the weaker requirement must just be what you get when you have any kind of function at all.

It’s all at

http://en.wikipedia.org/wiki/Bijection,_injection_and_surjection

Of course, the crafty teacher will operate a ‘Secret Santa’ system, by which every child is allocated a randomly selected other child, and then has to think of something positive about that child.

Thank you for your help!There is obviously a lot to know about this. I think you made some good points in Features also.