1. GENERAL REMARKS

(1) If you want to justify why, for example, a relation satisfies certain property, you must convince the reader that the property holds for every single element of the relation. More generally, if you want to justify why certain kinds of objects have a certain property, you have to give a proof, i.e., a convincing argument why the property holds for all possible such objects.

(2) If you want to justify why, for example, a relation does not satisfy a certain property, it is enough to give one example that shows that the property fails. Such an example is called a counterexample. More generally, if you want to justify why a particular object does not satisfy a particular property, it is enough to give a counterexample. Note that the counterexample has to be explicit.

(3) In mathematical problems, if you are asked to show something, that is the same as being asked to prove something, which means that you have to give a complete justification. I know that you may not be used to proving things yet, but that’s okay. We will learn this as we go. For now, remember that it is your job to convince the reader who is reading your solution, so you have to try your best to explain your answers.

(4) Important: If you are having trouble with any of the points mentioned above, come and discuss with me in office hour. It is part of my job to help you understand this stuff, so please use my time!

2. PROBLEMS

(1) Let R and T both be relations on a set S . For each statement below, either justify it or give a coun- terexample.

(a) If R and T are symmetric, then R ∪ T is symmetric.

(b) If R and T are transitive, then R ∪ T is transitive.

(2) Consider the following graphs.  For each one, write down which of the following properties are satisfied by the relation represented by the graph: reflexivity, symmetry, anti-symmetry, transitivity, being a function. You do not have to justify your answers, but you should think about the justifications instead of guessing.

(a)

(b)

Date: Due on 6 August 2021 at 11:59pm.