Hello there's a question where you need to find the total possible injective and surjective possibilities. Could someone verify my answer? I'm almost certain of answer a, but want to know if b is also correct?

If you know if its true can you reply, it would be very helpful. If it's wrong could you explain why?

Thanks in advance


assume |X|=2 and |Y|=2016

a) How many injective functions are there from X to Y? My answer = $\frac{2016!}{2014!}$

b) How many surjective functions are there from Y to X? My answer = $2^{2016}-2$

  • $\begingroup$ Whether or not these answers are right, you don't seem to have learned much mathematics. Solving a problem is more than "finding a formula" and asking whether it's right. If you edit the question to tell us why you think these formulas count what you are interested in, perhaps we can help. Note: that quotient of factorials simplifies a lot. $\endgroup$ Jan 9 at 16:39
  • $\begingroup$ This tutorial explains how to typeset mathematics on this site. $\endgroup$ Jan 9 at 16:44

Your answer for injections is correct.

Your answer for surjections is incorrect. In total we have $2^{2016}$ functions. However the only way a function is non-surjective in this case is if both of the elements in $X$ are mapped to the same element in $Y$, and there are $2016$ elements in $Y$.

So there are actually $2^{2016}-2016$ injections.

  • $\begingroup$ A wow indeed, thank you so much :) $\endgroup$
    – TheCreator
    Jan 9 at 16:53
  • $\begingroup$ You are confused somehow, the correct answer is $2^{2016}-2$. You need to subtract the two constant maps. $\endgroup$ Jan 9 at 17:26
  • $\begingroup$ You talk about the elements of X being mapped to elements of Y, but it should be the other way around. $\endgroup$ Jan 9 at 17:28

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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