Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I am a bit confused on the definition of a function and an injection. I can't seem to distinguish the two.

The definition of a function is: if $(x_1,y_1)\in f, (x_1,y_2)\in f$ then $y_1 = y_2$, whereas an injection is: if $(a,b)$ and $(a,',b)$ are elements of $F$, then $a=a'$; which in other words just means the domain of $f$ maps into distinct elements of $b$.

My question is, isn't the two definitions very similar? For example: $D = \{(x,y):|x|+|y| = 1\}$, is not a function because $y_1 \ne y_2$, but doesn't that mean it is not an injection because it doesn't map the domain into distinct elements which seems to me to be the precise definition of what it is to be a function? What is the difference between being a function and being injective?

share|cite|improve this question
up vote 1 down vote accepted

The relation between people and names is a function, because every person has a name, but it's not injective, because two different people can have the same name.

The relation between names and people is not a function, because one name can correspond to more than one person.

share|cite|improve this answer
Thanks Gerry! So, essentially I should concentrate on the range to tell if it is an injection? And similarly, I should concentrate on the domain to tell if it is a function? – Q.matin Feb 7 '13 at 5:25
Pretty much. Note that if it's not a function, it doesn't make sense to ask whether it's an injection. – Gerry Myerson Feb 7 '13 at 5:55
Thanks a lot !! – Q.matin Feb 7 '13 at 6:02

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.