List one of three properties of an equivalence relations

Question

I have the following question and answers:

Let $A = \{1,2,3,4\}$. Write down $R \subseteq A x A$ which has one of the three properties of an equivalence relation (3 cases).

The official answers for the above are (taken from textbook):

Case 1: Only reflexive

$R_1 = \{(1,1), (2,2), (3,3), (4,4), (1,2), (2,3)\}$

My question: Is it necessary to include (1,2) and (2,3) here? Can I not include other pairs such as (1,4) and (2,4) in order to ensure this is just reflexive?

Case 2: Only symmetric

$R_2 = \{(1,2), (2,1)\}$

My question: Do I really have to use (1,2) and (2,1) here? Can I not use (3,1), (1,3) which is inside the A x A?

Case 3: Only transitive

$R_3 = \{(1,1), (1,3)\}$

My question: Similar to symmetric, must do I really have to use (1,1) and (1,3)? Can I show this is transitive using (1,2) and (2,3) - since (1,3) is also in AxA.

The reason why I am confused is because I thought the available pairs for this AxA is

Hence I should be able to pick any pairs there as long as it fulfill the requirement of the question?

Answer 1

Change the names of the elements of $A$. Say $A=\{a,b,c,d\}$. For instance, the answer to case 2 now is $\{(a,b),(b,a)\}$. It is clear that the change of names do not change essentially anything?

Now change again:

$$A=\{1,3,2,4\}.$$

How is now changed case 2? Has this solved your question to case 2?

This not directly solves your other questions, but I hope make you walk for the right path.

Answer 2

Case $1)$ is only reflexive. It's not necessary to use $(1,2)$ and $(2,3)$. You can replace them by $(1,3)$ and $(3,4)$, say.

Etc.

Case $2)$. There's more than one way to do it. There are plenty of other choices available. But whenever you have $(a,b)$, you need $(b,a)$.

Case $3)$. Again, there's more than one way to do it. Yes, you can pick different pairs. As long as whenever you have $(a,b)$ and $(b,c)$, you have $(a,c)$. And also something to contradict reflexivity and symmetry. That is, you can not have all four of $(1,1),(2,2),(3,3),(4,4)$, or else it will be reflexive, for instance.

Of course you can use other pairs, as long as you are careful.