# Very Basic Relational Properties

$R = \{ (1,3),(2,3), (3,4) \}$ It says this $R$ is anti-symmetric but how, $a$ does not equal $b$

$R = \{ (1,1),(2,2),(3,3),(4,4) \}$ It says this is Reflexive (yup) Symmetric (?), Anti-Symmetric (?), and Transitive (?), and I was wondering how this makes sense.

If you draw a graph to represent these I don't know how its true. Maybe the solutions is messed?

Btw, it's set of relations on the set $A = \{ 1,2,3,4 \}$

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Anti-symmetric just means you can't have both $(a,b)$ and $(b,a)$, right? –  Gerry Myerson Nov 12 '12 at 0:25
No, it's if u have both (a,b)and(b,a) then => a=b –  Aaron Nov 12 '12 at 0:28
OK, then what's the difficulty? There are no $a$ and $b$ such that $R$ has both $(a,b)$ and $(b,a)$, so the "then" part never eventuates, right? –  Gerry Myerson Nov 12 '12 at 1:04
Hey you are right! Sorry!! –  Aaron Nov 12 '12 at 1:05

Edit: Now that you have changed the first problem, it is actually anti-symmetric. Why? Because being anti-symmetric means obeying an if-statement that says if you have $(x,y)$ and $(y,x)$, then $x = y$. Here, you don't have anything satisfying the if part of this conditional; thus, we say the if-then statement (vacuously) holds.
As for the second example, this is certainly symmetric and transitive: both of those properties are if-then statements, and here the "if" part can only be satisfied by ordered pairs of the form $(n,n)$. In fact, this relation is a formal way of writing what we would normally represent with "$=$", that is, equality in the standard sense of arithmetic, which you probably know is an equivalence relation.