0
$\begingroup$

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

  • 0
    Hey you are right! Sorry!!2012-11-12

1 Answers 1

2

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.

Thinking of the second relation as equality, hopefully you can see why anti-symmetry holds too, though it does so in a somewhat vacuous manner (as in the first example).

  • 0
    Hey, I have one more question, is (2,3),(3,2) is this transivitive.2012-11-12