Anti symmetrical relation

Question

Currently learning about symmetrical and anti symmetrical relations. Been working on assignments but I cant seem to bend my head around this one, especially because I can't understand the solution.

What I thought an anti symmetrical relation is: a pair like $(x,y)$ that also exists as $(y,x)$ with $x = y.$

The question is: describe the relation of the following collection.

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

Answer: anti symmetrical.

Why is this anti symmetrical?

Answer 1

Suppose that $R$ is a relation on a set $A$. When we say that $R$ is symmetric, we mean that if one of the pairs $\langle a,b\rangle$ and $\langle b,a\rangle$ is in $R$, then so is the other. When we say that $R$ is antisymmetric, we mean that if $a\ne b$, and one of the pairs $\langle a,b\rangle$ and $\langle b,a\rangle$ is in $R$, then the other one is not in $R$.

This is the case with your relation

$$R=\{\langle 1,2\rangle,\langle 1,3\rangle,\langle 1,4\rangle,\langle 2,3\rangle,\langle 2,4\rangle,\langle 3,4\rangle\}\;:$$

$\langle 1,2\rangle$ is in it, but $\langle 2,1\rangle$ is not; $\langle 2,3\rangle$ is in it, but $\langle 3,2\rangle$ is not; and so on for every possible pair of distinct elements of $\{1,2,3,4\}$.

Note that for both symmetry and antisymmetry it’s perfectly fine to have neither pair in the relation. The relation

$$\{\langle 1,2\rangle,\langle 2,1\rangle,\langle 1,3\rangle,\langle 3,1\rangle\}$$

is symmetric even though it has neither of the pairs $\langle 2,3\rangle$ and $\langle 3,2\rangle$, and the relation

$$\{\langle 1,2\rangle,\langle 3,1\rangle\}$$

is antisymmetric even though it has neither of the pairs $\langle 2,3\rangle$ and $\langle 3,2\rangle$.

Answer 2

For the definition of an antisymmetric relation see here. It is unclear to me if this is what you thought. In any case, it is easy to check that your example is antisymmetric.

For example, antisymmetric means that $R$ cannot contain both the pair $(1,2)$ and the pair $(2,1)$ without $1$ and $2$ being equal. This is not the case in the example.

Answer 3

An antisymmetric relation $R$ is such that if $(x, y)$ and $(y, x)$ are both in $R$, then this implies that $x = y$.

It your case, the hypothesis is never satisfied, so the conclusion doesn't matter for any elements. That is, it is vacuously true that the relation is antisymmetric.

Another way to look at the property of antisymmetry is that a relation $R$ is antisymmetric on a set $S$ UNLESS there exists $x, y \in S$ such that $(x, y), (y, x) \in R$, but $x\neq y$. Then, and only then, is the relation not antisymmetric.