The hint for Question 4 may be helpful for Question 2, so I have put it first.

**Question 4.** This question asks you to find surjective functions and such that the function defined by is not surjective. One way to get a function that is not surjective is to take a constant function which takes only one value. For example, the function defined by for all . If you could arrange that and were surjective, and

for all , then you would be done.

**Question 2.** Your job is to define a function such that is neither injective nor surjective. By definition of surjective, is surjective if for each , there exists at least one such that . So to make sure that is not surjective, you need to make sure that there is some which is not equal to any for . Given this guide, try to define . If you find that with your definition, is also not injective, you have a suitable example.

**Question 6.** We haven’t covered relations yet in lectures. But you should be able to make an attempt at this and Question 7 by reading pages 35 and 36 of the lecture notes, issued in Monday’s lecture.

Here is a detailed solution to (6c). We define a relation on the set of people in a lecture room by if can see the eyes of . The relation is *reflexive* if for all we have , i.e. if everyone in the lecture room can see their own eyes. I think we can safely say this is false. (Not everyone can be looking in a mirror at once.) It is *symmetric* if

i.e., if and see the eyes of then can see the eyes of . This is true (at least if we make the kind assumption that everyone’s eyes are always open in a lecture). Finally, the relation is *transitive* if

In words, this says that if can see the eyes of , and can see the eyes of , then can see the eyes of . This is false: for example take to be the lecturer and and to be two people facing the lecturer. So is symmetric but not reflexive or transitive.

**Question 7.** Here are some examples for (a). We have since and

So you should draw an arrow from to . Also , since

so there should also be an arrow from to . Similarly there should be arrows from to , from to . There should also be a loop from to , since . In the diagram for Example 8.2, I used a single line with two arrow-heads instead of two separate arrows.

**Question 8.** For (c), you should use the formula for the inverse of a composition given at the top of page 34 of the lecture notes:

Applying this to we get

You should now use your formula from (b) for and then your formula from (a) for to turn the right-hand side of the equation above into an explicit formula for .