Identifying if a relation is reflexive, symmetric and/or transitive I am totally lost on how to identify if a relation is the above.
The only thing I know if you have a matrix, and is diagonally symmetric, then it is symmetric, but I do not know why.
Could someone elaborate a little bit using (sorry for my ignorance) those maps with circles and arrows? Say if you have in a set (a,b) (b,a) (a,c) (c,a) you would have a, b and c inside a circle and then you would draw arrows from a to b, b to a, a to itself, a to c... etc.. 
I also have a problem that says:
Let the relation R be the set of all the students in a group be defined as follows:
For (x,y) ∈ R if and only if x & y have different majors
In this case, what would it be? (reflexive, symmetric and/or transitive)
Thank you for your time and advice
 A: A relation is called symmetric  when "$a$ is in relation to $b$" if and only if "$b$ is in relation to $a$". If you want to visualize this using arrows it would mean that if you have an arrow between two points than you also have an arrow in the converse direction. 
The relation you state has this property if Jack and Jill have different majors then Jill and Jack also have different majors.
Put differently, this is a relation where the order of the two things put into relation or not is irrelevant. Relations based on things sharing (or not sharing) a certain property are typically symmetric. Note that if your relation would be "have the same major" it would also be symmetric.
A relation is called reflexive when "$a$ is in relation with itself." Equality is a prime example. Or, having a certain property. Using arrows this would mean there is an arrow from each element to itself, like a loop.
Your relation is not reflexive. Of course, Jill has the same major as Jill (assuming it is the same Jill), so it is not true that Jill and Jill have a different major. However, if the relation was based on "have the same major" it would be reflexive.
A relation is called transitive when "$a$ is in relation to $b$" and "$b$ is in relation to $c$" implies that "$a$ is in relation to $c$." Comparing by size is a good example here: if Jill is taller than Jack, and Jack is taller than John, then you certainly can infer Jill is taller than John.  
Using arrows this would mean if you can connect two points with two arrows passing through another point then you also have a direct connection. 
You relation is not transitive. If the major of Jill is math, and the major of Jack is philosophy, and the major  of Jane is math. 
Then Jill and Jack are in relation, as they have different majors, and Jack and Jane are in relation as they have different majors. But Jill and Jane are not in relation as they have the same major.
A: In a directed graph, you have the following properties:


*

*The relation is reflexive, iff there is an arrow for each object to itself.

*The relation is symmetric, iff there is for each arrow an arrow in the opposite direction, i.e. whenever you have an arrow $a \rightarrow b$ you also have an arrow $b \rightarrow a$

*The relation is transitive, iff you can always "take a shortcut", i.e. whenever you have an arrow $a \rightarrow b$ and an arrow $b \rightarrow c$ you also have a direct arrow $a \rightarrow c$ (a "shortcut" from $a$ to $c$)


I hope the above descriptions help you...
