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What are some mnemonics to help one remember that Injection = One-to-one and Surjection = Onto? The only thing I can think of is 1njection = 1-1.

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I blame bad terminology. Until someone explains why these are called what they are called, I blame bad terminology. – Sabyasachi Apr 6 '14 at 14:20
@Sabyasachi I think it’s great terminology. See the answer by fgp. – k.stm Apr 6 '14 at 14:27
It is not bad terminology. One only needs to know one Latin language. – Sergio Parreiras Apr 6 '14 at 14:29
@k.stm Not latin again. -.- at least now it makes sense. Btw I already saw that answer and upvoted. – Sabyasachi Apr 6 '14 at 14:34
Practice any concept enough, and the terminology settles down in your mind. – Sawarnik Apr 6 '14 at 17:08

An injection $A \to B$ maps $A$ into $B$, i.e. it allows you to find a copy of $A$ inside $B$.

A surjection $A \to B$ maps $A$ over $B$, in the sense that the image covers the whole of $B$. The syllable "sur" has latin origin, and means "over" or "above", as for example in the word "surplus" or "survey".

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Addendum: To be clear, “sur” is French from Latin “super”. – k.stm Apr 6 '14 at 14:26
@k.stm: That's super! Err, I mean... That's sur..! :-P – Asaf Karagila Apr 7 '14 at 3:30

An injection $A\to B$ provides a correspondence between $A$ and some subset of $B$ -- that, is an INjection points to a copy of $A$ INside $B$.

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I'd replace "provides a correspondence" to "is a one-to-one correspondence". And the "copy of $A$" confounds me. I'd just say: "between $A$ and a some subset of $B$ -- that is, a set that is INside $B$. – leonbloy Apr 6 '14 at 20:01

The way I remember it is that when you get a flu shot your entire body doesn't turn into a giant flu virus, because the needle is smaller than your arm is. Then you can easily remember surjection as "the other one".

Another one is that in-jections are in-ferior and su-rjections are su-perior.

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This also makes sense in that if $f:A\rightarrow B$ is injective, then $A\le B$ and if it is surjective then $A\ge B$. – Thomas Ahle Aug 21 '14 at 8:56

The best way to remember is to only remember one, then by elimination you know the other.

I choose to remember injective as follows:

Injections cure things, and you have one injection for one cure. I.e. one to one.

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This doesn't make much sense since there could be many injections that cure the same disease. – Istvan Chung Apr 7 '14 at 2:10
Yes but you generally just get one. – ellya Apr 7 '14 at 6:04

Take a look at this picture:

This function is NOT injection, because two arrows point into single point in that picture.

Now imagine injections at the doctor. Injections usually hurt and you, sure as hell, woudln't want anyone to stick that injection into the same point on your body multiple times.

So that's why injective functions cannot have multiple arrows pointing into the same point (value)


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