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.

Answer 1

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 $B$.

Answer 2

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".

Answer 3

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.

Answer 4

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)

:)

Answer 5

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.

Answer 6

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.