Q:

Is this antisymmetric? Why?

This is from a past paper exam I am revising. Please can someone explain if this is antisymmetric or not. According to the answer it says it isn't, but I can't for the life of me understand why.
Consider the set N
+ of all non-empty finite sequences of natural numbers
– for example, [1, 1], [1, 2, 1] and [2, 0, 2, 0, 0] are all elements of N
+. Let R
be the binary relation over N
+ defined by (s, t) ∈ R if and only if s1 ≤ t1
where s1 and t1 are the first numbers of the sequences s and t, respectively.
For example, ([1, 1], [1, 2, 1]) ∈ R but ([2, 0, 2, 0, 0], [1, 1]) ∈/ R.

A:

It is not antisymmetric because, for instance, $[1\ \ 1]\mathrel R[1\ \ 2]$ and $[1\ \ 2]\mathrel R[1\ \ 1]$ , but you don't have $[1\ \ 1]=[1\ \ 2]$ .

