Injection vs. Surjection: Mnemonic to remember which is which

Question

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.

I blame bad terminology. Until someone explains why these are called what they are called, I blame bad terminology. — Guy, Apr 06 '14 at 14:20
@Sabyasachi I think it’s great terminology. See the answer by fgp. — k.stm, Apr 06 '14 at 14:27
It is not bad terminology. One only needs to know one Latin language. — Sergio Parreiras, Apr 06 '14 at 14:29
@k.stm Not latin again. -.- at least now it makes sense. Btw I already saw that answer and upvoted. — Guy, Apr 06 '14 at 14:34
@Sabyasachi “terminology” is Latin! : – D (and Ancient Greek, of course.) — k.stm, Apr 06 '14 at 14:37
Practice any concept enough, and the terminology settles down in your mind. — Sawarnik, Apr 06 '14 at 17:08

score 28 · Answer 1 · edited Apr 06 '14 at 14:34

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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".

edited Apr 06 '14 at 14:34

Jack M

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answered Apr 06 '14 at 14:24

fgp

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Addendum: To be clear, “sur” is French from Latin “super”. – k.stm Apr 06 '14 at 14:26
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@k.stm: That's super! Err, I mean... That's sur..! :-P – Asaf Karagila Apr 07 '14 at 03:30

score 13 · Answer 2 · edited Aug 13 '19 at 05:05

Take a look at this picture (from Wikipedia):

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)

:)

score 5 · Answer 3 · answered Apr 06 '14 at 19:28

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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.

answered Apr 06 '14 at 19:28

Hovercouch

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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 08:56

score 3 · Answer 4 · answered Apr 06 '14 at 14:22

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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 $B$.

answered Apr 06 '14 at 14:22

hmakholm left over Monica

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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 06 '14 at 20:01

score 1 · Answer 5 · answered May 02 '23 at 13:14

The way I remember them:

An INjective function is INvertible (not strictly true, but it also helps me remember how to prove injectivity): Schematically, $f^{-1}(f(x) = f(y)) \implies x = y$
A SURjective function is SURe to hit all the elements of the target.

score 1 · Answer 6 · answered Apr 06 '14 at 14:28

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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.

answered Apr 06 '14 at 14:28

Ellya

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This doesn't make much sense since there could be many injections that cure the same disease. – lily Apr 07 '14 at 02:10
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Yes but you generally just get one. – Ellya Apr 07 '14 at 06:04

Injection vs. Surjection: Mnemonic to remember which is which

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