Chapter 35 Properties of relations
Symbolization keys allow us to assign twoplace predicate symbols like ‘$R(x,y)$’ to English predicates with two gaps, e.g., ‘blank${}_{x}$ is older than blank${}_{y}$’. This allows us to symbolize arguments such as:

Rudolf is older than Kurt.

Kurt is older than Julia.

∴
Rudolf is older than Julia.
This argument is valid, but its symbolization in FOL,

$R(r,k)$

$R(k,j)$

∴
$R(r,j)$
is not. That’s because the validity of the argument depends on a property of the relation ‘$x$ is older than $y$’, namely that if some person $x$ is older than a person $y$, and $y$ is also older than $z$, then $x$ must be older than $z$. This property of ‘older than’ is called transitivity . We can symbolize this property itself in FOL as:
Whenever the validity of an argument only depends on a property of a relation involved, and this property can be symbolized in FOL, we can add this symbolization to the argument and obtain an argument that is in fact valid in FOL:

$\forall x\forall y\forall z((R(x,y)\wedge R(y,z))\to R(x,z))$

$R(r,k)$

$R(k,j)$

∴
$R(r,j)$
Properties of relations such as transitivity are important concepts especially in applications of logic in the sciences. For instance, order relations between numbers such as $$ and $\le $ (and also $>$ and $\ge $) are transitive. The identity relation $=$ is also transitive.
There are other properties of relations that are important enough to have names, and often show up in applications. Here are some:

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A relation $\colorbox[rgb]{0.992156862745098,0.956862745098039,0.980392156862745}{$R$}$ is transitive iff it is the case that whenever $\colorbox[rgb]{0.992156862745098,0.956862745098039,0.980392156862745}{$R$}\colorbox[rgb]{0.992156862745098,0.956862745098039,0.980392156862745}{$$}\colorbox[rgb]{0.992156862745098,0.956862745098039,0.980392156862745}{$($}\colorbox[rgb]{0.992156862745098,0.956862745098039,0.980392156862745}{$x$}\colorbox[rgb]{0.992156862745098,0.956862745098039,0.980392156862745}{$,$}\colorbox[rgb]{0.992156862745098,0.956862745098039,0.980392156862745}{$y$}\colorbox[rgb]{0.992156862745098,0.956862745098039,0.980392156862745}{$)$}$ and $\colorbox[rgb]{0.992156862745098,0.956862745098039,0.980392156862745}{$R$}\colorbox[rgb]{0.992156862745098,0.956862745098039,0.980392156862745}{$$}\colorbox[rgb]{0.992156862745098,0.956862745098039,0.980392156862745}{$($}\colorbox[rgb]{0.992156862745098,0.956862745098039,0.980392156862745}{$y$}\colorbox[rgb]{0.992156862745098,0.956862745098039,0.980392156862745}{$,$}\colorbox[rgb]{0.992156862745098,0.956862745098039,0.980392156862745}{$z$}\colorbox[rgb]{0.992156862745098,0.956862745098039,0.980392156862745}{$)$}$ then also $\colorbox[rgb]{0.992156862745098,0.956862745098039,0.980392156862745}{$R$}\colorbox[rgb]{0.992156862745098,0.956862745098039,0.980392156862745}{$$}\colorbox[rgb]{0.992156862745098,0.956862745098039,0.980392156862745}{$($}\colorbox[rgb]{0.992156862745098,0.956862745098039,0.980392156862745}{$x$}\colorbox[rgb]{0.992156862745098,0.956862745098039,0.980392156862745}{$,$}\colorbox[rgb]{0.992156862745098,0.956862745098039,0.980392156862745}{$z$}\colorbox[rgb]{0.992156862745098,0.956862745098039,0.980392156862745}{$)$}$.

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A relation $\colorbox[rgb]{0.992156862745098,0.956862745098039,0.980392156862745}{$R$}$ is reflexive iff for any $\colorbox[rgb]{0.992156862745098,0.956862745098039,0.980392156862745}{$x$}$, $\colorbox[rgb]{0.992156862745098,0.956862745098039,0.980392156862745}{$R$}\colorbox[rgb]{0.992156862745098,0.956862745098039,0.980392156862745}{$$}\colorbox[rgb]{0.992156862745098,0.956862745098039,0.980392156862745}{$($}\colorbox[rgb]{0.992156862745098,0.956862745098039,0.980392156862745}{$x$}\colorbox[rgb]{0.992156862745098,0.956862745098039,0.980392156862745}{$,$}\colorbox[rgb]{0.992156862745098,0.956862745098039,0.980392156862745}{$x$}\colorbox[rgb]{0.992156862745098,0.956862745098039,0.980392156862745}{$)$}$ holds.

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A relation $\colorbox[rgb]{0.992156862745098,0.956862745098039,0.980392156862745}{$R$}$ is symmetric iff whenever $\colorbox[rgb]{0.992156862745098,0.956862745098039,0.980392156862745}{$R$}\colorbox[rgb]{0.992156862745098,0.956862745098039,0.980392156862745}{$$}\colorbox[rgb]{0.992156862745098,0.956862745098039,0.980392156862745}{$($}\colorbox[rgb]{0.992156862745098,0.956862745098039,0.980392156862745}{$x$}\colorbox[rgb]{0.992156862745098,0.956862745098039,0.980392156862745}{$,$}\colorbox[rgb]{0.992156862745098,0.956862745098039,0.980392156862745}{$y$}\colorbox[rgb]{0.992156862745098,0.956862745098039,0.980392156862745}{$)$}$ holds, so does $\colorbox[rgb]{0.992156862745098,0.956862745098039,0.980392156862745}{$R$}\colorbox[rgb]{0.992156862745098,0.956862745098039,0.980392156862745}{$$}\colorbox[rgb]{0.992156862745098,0.956862745098039,0.980392156862745}{$($}\colorbox[rgb]{0.992156862745098,0.956862745098039,0.980392156862745}{$y$}\colorbox[rgb]{0.992156862745098,0.956862745098039,0.980392156862745}{$,$}\colorbox[rgb]{0.992156862745098,0.956862745098039,0.980392156862745}{$x$}\colorbox[rgb]{0.992156862745098,0.956862745098039,0.980392156862745}{$)$}$.

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A relation $\colorbox[rgb]{0.992156862745098,0.956862745098039,0.980392156862745}{$R$}$ is antisymmetric iff for no two different $\colorbox[rgb]{0.992156862745098,0.956862745098039,0.980392156862745}{$x$}$ and $\colorbox[rgb]{0.992156862745098,0.956862745098039,0.980392156862745}{$y$}$, $\colorbox[rgb]{0.992156862745098,0.956862745098039,0.980392156862745}{$R$}\colorbox[rgb]{0.992156862745098,0.956862745098039,0.980392156862745}{$$}\colorbox[rgb]{0.992156862745098,0.956862745098039,0.980392156862745}{$($}\colorbox[rgb]{0.992156862745098,0.956862745098039,0.980392156862745}{$x$}\colorbox[rgb]{0.992156862745098,0.956862745098039,0.980392156862745}{$,$}\colorbox[rgb]{0.992156862745098,0.956862745098039,0.980392156862745}{$y$}\colorbox[rgb]{0.992156862745098,0.956862745098039,0.980392156862745}{$)$}$ and $\colorbox[rgb]{0.992156862745098,0.956862745098039,0.980392156862745}{$R$}\colorbox[rgb]{0.992156862745098,0.956862745098039,0.980392156862745}{$$}\colorbox[rgb]{0.992156862745098,0.956862745098039,0.980392156862745}{$($}\colorbox[rgb]{0.992156862745098,0.956862745098039,0.980392156862745}{$y$}\colorbox[rgb]{0.992156862745098,0.956862745098039,0.980392156862745}{$,$}\colorbox[rgb]{0.992156862745098,0.956862745098039,0.980392156862745}{$x$}\colorbox[rgb]{0.992156862745098,0.956862745098039,0.980392156862745}{$)$}$ both hold.
We have already seen that transitivity can be symbolized in FOL. The others can, too:

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$R$ is reflexive: $\forall xR(x,x)$

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$R$ is symmetric:
$$\forall x\forall y(R(x,y)\to R(y,z))$$ 
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$R$ is antisymmetric:
$$\mathrm{\neg}\exists x\exists y((R(x,y)\wedge R(y,x))\wedge \mathrm{\neg}x=y)$$or, equivalently,
$$\forall x\forall y((R(x,y)\wedge R(y,x))\to x=y)$$
Relations expressing an equivalence in some respect are reflexive, e.g., ‘is the same age as’, ‘is as tall as’, and the most stringent equivalence of them all, identity $=$. They are also symmetric: e.g., whenever $x$ is as tall as $y$, then $y$ is as tall as $x$. In fact, relations that are reflexive, symmetric, and transitive are called equivalence relations .
Equivalences aren’t the only symmetric relations. For instance, ‘is a sibling of’ is symmetric, but it is not reflexive (as no one is their own sibling).
Relations that are reflexive, transitive, and antisymmetric are called partial orders . For instance, $\le $ and $\ge $ are partial orders. The relation ‘is no older than’, by contrast, is reflexive and transitive, but not antisymmetric. (Two different people can be of the same age, and so neither is older than the other.)
Practice exercises
A. Give examples of relations with the following properties:

1.
Reflexive but not symmetric

2.
Symmetric but not transitive

3.
Transitive, symmetric, but not reflexive
B. Show that a relation can be both symmetric and antisymmetric by giving an interpretation that makes both of the following sentences true:
$\forall x\forall y(R(x,y)\to R(y,z))$  
$\forall x\forall y((R(x,y)\wedge R(y,x))\to x=y)$ 