# How many injective and surjective functions are there from $A$ to $B$? [closed]

This is an old practice exam question in my calculus class and I would love to understand it before my exam, so any help would be great!

Consider the sets $A = \{a,b\}$ and $B=\{a,c,d,e,f\}$.

a) How many functions are there from $A$ to $B$?

b) How many injective functions are there from $A$ to $B$?

c) How many surjective functions are there from $A$ to $B$?

I know what injective and surjective means, but I have only applied it in linear algebra and I'm not sure how to do these.

## closed as off-topic by Namaste, B. Mehta, N. F. Taussig, GNUSupporter 8964民主女神 地下教會, qwrApr 14 '18 at 19:06

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• How many ways can $a\in A$ be mapped to one and only one of the five elements in $B$? How many ways can $b\in A$ then be mapped to one and only one of the elements in $B$? – Namaste Apr 14 '18 at 18:53
• Now, we have five ways to map $a \in A$ to one of the elements in $B$. For a mapping to be injective, $b \in A$ can be mapped to any of four elements not mapped to the element in $B$ that $a$ was mapped to. No function from $A\to B$ can be surjective. Why? Recall the definition of a function. – Namaste Apr 14 '18 at 19:04

Consider the sets A={a,b} and B={a,c,d,e,f}.

a) How many functions are there from A to B?

The answer is $5^2 =25$ because you have $5$ choices for each $a$ or $b.$

b) How many injective functions are there from A to B?

The answer is $5\times 4 =20$.you have $5$ choices for $a$ and only $4$ choices for $b$

c) How many surjective functions are there from A to B?

The answer is $0$. You can not cover $5$ elements with just $a$ and $b$

Hints: a) Think about defining a function from $A \to B$. How many functions can you think of? How many choices do you need to make to define such a function?

b) Thinking about the functions above - how many are not injective? How can such a function fail to be injective? So, how many injections are there?

c) I'll leave to you - but think along similar lines as to b).

• Could the downvoter explain what they think could be improved about this answer? – B. Mehta Apr 14 '18 at 18:55
• No, as downvoting on SE is anonymous. Btw, I didn't downvote. – GNUSupporter 8964民主女神 地下教會 Apr 14 '18 at 18:59
• Sure, but I'm requesting the downvoter to clarify how they think I could improve this response. – B. Mehta Apr 14 '18 at 19:00