Relations. Are these sets reflexive, transitive, symmetric, etc.?

Question

I'm studying relations at the moment, but I can't solve this problem. Could you please give me a piece of advice? I really want to understand how to solve it. Thank you in advance!

Let R be a relation on a nonepty set X. Is this relation reflexive, transitive, symmetric, antisymmetric? Is it equivalence relation, partial order, total order? If it's an equivalence relation, identify the equivalence classes.

1. X = {1,2,3,4,5,6,7,8,9}, R = {(a,b)| a + b is even number},

2. X = $\mathbb R^2$, R = "symmetric with respect to x-axis",

3. X =$\mathbb R^2$, R = "is at the same distance form origin",

4. X = {all human beings}, R = "to be brother",

5. X = {all human beings}, R = "live in the same city".

Here are my thoughts on this so far (I may be really wrong, but anyway):

1. Reflexive and symmetric.
2. Symmetric.
3. Reflexive and symmetric.
4. Antisymmetric?
5. Also antisymmetric?

Answer 1

5) is symmetric (if x lives in same city as y, then y lives in same city as x).

5) is also reflexive (x lives in same city as x), but 4 is not (no one is brother of themselves .. so 4 is in fact irreflexive).

4) is not antisymmetric: x and y can be brothers of each other. Also not symmetric (x can be brother of y, but y can be sister of x)

Any thoughts on transitivity for these?

Answer 2

a and b live in the same city when there is some city in which both a and b live.

Joe is a farmer who lives on his farm outside of any city.
Joe is in X. There is no city in which both Joe and Joe live.

Thus 5 is not an equivalence relation because it is not reflexive.