Elementary Foundations: An introduction to topics in discrete mathematics

The distinct part of the definition is important, since if $a_1,a_2\in A$ are not distinct (i.e. $a_2=a_1$), then obviously both $a_{1}R{a}_2$ and $a_{2}R{a}_1$ can be simultaneously true because they are the same statement.

As Example 17.3.7 and Example 17.3.8 demonstrate, antisymmetry is not the opposite of symmetry. However, for a relation $R$ on set $A\text{,}$ we may think of symmetry and antisymmetry as being at opposite ends of a spectrum, measuring how often we have both $a_{1}R{a}_2$ and $a_{2}R{a}_1$ for $a_1\ne a_2\text{.}$

By definition, antisymmetry is when we never have both. On the other hand, symmetry is when we always have both or neither; that is, for every distinct pair $a_1,a_2\in A\text{,}$ we either have both $a_{1}R{a}_2$ and $a_{2}R{a}_1\text{,}$ or we have both $a_1\mathrm{R̸}{a}_2$ and $a_2\mathrm{R̸}{a}_1\text{.}$ However, a relation can fall between symmetry and antisymmetry on the spectrum, such as in Example 17.3.7, where we sometimes have both (e.g. both $aRb$ and $bRa$ for that example relation) and we also sometimes have only one (e.g. $aRc$ but $c\mathrm{R̸}a$ for that example relation).

The equality relation on a set is a special case that is both symmetric and antisymmetric. In fact, equality is essentially the only relation that is both symmetric and antisymmetric — see Exercise 17.6.22.

The first formulation for proving antisymmetry provided above can be thought of as just a different way to say that it is not possible to have both $a_{1}R{a}_2$ and $a_{2}R{a}_1$ for distinct elements $a_1,a_2\text{.}$ The second formulation essentially says that the only possible way to have both $a_{1}R{a}_2$ and $a_{2}R{a}_1$ is if $a_2=a_1\text{.}$