Real life example of relations with various combination of properties

Question

Attempted a set of questions as below:

Give examples of relations that are
1. asymmetric
2. reflexive, symmetric, but not transitive
3. antisymmetric, transitive, but not reflexive 
4. reflexive, transitive, but not antisymmetric (equivalence)

I solved the first three questions but I couldn't solve the last one.

My answers for the first two:
1). x is a father of y
2). x lives within one mile of y
3). x is an ancestor of y

Answer 1

1. Asymmetric -- x is a father of y

This is a perfectly fine example of an asymmetric relation. After all, if John is Paul's father, Paul is assuredly not John's father in turn.

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2. reflexive, symmetric, but not transitive -- x lives within one mile of y

Perfectly valid as well:

• Any person lives within a mile of themselves (namely zero distance), so it is reflexive.
• If one person lives within a mile of another, that person consequently lives within a mile of the first, so it's symmetric.
• It is not ensured that if Paul lives within a mile of John, who lives within a mile of Martha, that Paul is within a mile of Martha. For instance, if Paul and Martha are two miles apart, and John is exactly between the two, we see a lack of transitivity.

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3. antisymmetric, transitive, but not reflexive -- x is an ancestor of y

This one works as well, but I wouldn't say it's a stellar example considering how antisymmetry works here.

• If X is Y's ancestor, in the typical usage, Y is never X's ancestor. Thus we have antisymmetry: since we never have both, we have no worries about the conclusion. (Recall: antisymmetry for a relation $R$ is that $xRy$ and $yRx$, if both true, always imply $x=y$. Here, we would never have one of the two premises, so $x=y$ doesn't matter. So it works but it doesn't necessarily scream a good example. But that's a matter of opinion.)

• Of course, if X is Y's ancestor, who is in turn Z's ancestor, Z is an ancestor of X, so we have transitivity.

• No one is their own ancestor, so this relation is not reflexive.

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4. reflexive, transitive, but not antisymmetric (equivalence)

First, I want to state despite some nudges in the comments in this way -- an equivalence relation is explicitly reflexive, transitive, and symmetric. "Not antisymmetric" is not the same as "symmetric." You can have relations which are both symmetric and antisymmetric, and relations which are neither.

For instance, on the set $\{1,\cdots,10\}$, let us define a few relations:

• $R_1 = \{(1,1)\}$
• $R_2 = \{(1,10),(10,1),(10,2)\}$
• $R_3 = \{(2,5),(5,2),(3,3)\}$
• $R_4 = \{(4,8)\}$

We analyze symmetry and antisymmetry as so:

• $R_1$ is symmetric and antisymmetric outright. $1R_11$ is true, and in turn $1R_11$ is true as well, so you have symmetry. But this same logic alongside the fact $1=1$ gives antisymmetry too. (In fact, any relation which consists solely of a set of pairs of the form $(x,x)$ is both antisymmetric and symmetric.)

• $R_2$ is neither symmetric nor antisymmetric. $1R_210$ and $10R_21$, but $10 \ne 1$, so we don't have antisymmetric. $2R_210$ is true, but we don't have the reverse, so no symmetry.

• $R_3$ is symmetric, but not antisymmetric. $2R_35$ and $5R_32$ but $2 \ne 5$, so no antisymmetry. However, whenever $(x,y) \in R_3$ we have that $(y,x) \in R_3$ too, so we have symmetry.

• $R_4$ is antisymmetric, but not symmetric. We never have both premises of $xR_4y$ and $yR_4x$, so we already see antisymmetry. However, $4R_48$ is true where $8R_44$ isn't, so no symmetry.

For these reasons, it is not right to interpret antisymmetry as the opposite of symmetry, but rather antisymmetry is as far as one could get from symmetry - whenever symmetry would hold it just means the two related objects are the same thing.

Some of these notions are easier to see and understand in the sense of representing relations as matrices. You can see this briefly discussed in my answer here where specifically symmetry and antisymmetry are concerned, since that seems to be the crux of this issue.

So, back to the comments and their suggestions -- would an equivalence relation work as an example? Sure, since symmetry doesn't negate antisymmetry. A trivial example of this is, say, a relation where everything is related to everything. Then:

• Reflexivity: Everything is related to itself
• Symmetry: If $xRy$ then $yRx$ (since each is related to everything)
• Transitivity: If $xRy$ and $yRz$ is true (always is), then $xRz$ is true (because $x,y,z$ are related to everything)
• Not Antisymmetric: Assume distinct $x,y$. Then $xRy$ and $yRx$ is always true since, again, everything is related, but $x \ne y$ on assumption.

The first three give equivalence, and the first two and the last are the properties you desire. But again, equivalence relations are a bit too strong here, insofar as having properties not strictly necessary, so perhaps an example of a relation where you have reflexivity and transitivity, but not symmetry or antisymmetry, would help.

Let us consider a relation on the set of people in the world, and define a relation $\sim$ by

$$x \sim y \iff \text{age}(x) \ge \text{age}(y)$$

In words, a person $X$ is related to a person $Y$ under $\sim$ if $X$ is at least as old as $Y$. (For simplicity, age is measured only in years, rounded down as necessary.) Let's verify:

• Reflexive: $\text{age}(x) = \text{age}(x)$, and thus $\text{age}(x) \ge \text{age}(x)$, so $x \sim x$ obviously.
• Transitive: Let $\text{age}(x) \ge \text{age}(y)$ and $\text{age}(y) \ge \text{age}(z)$. That is, let $x \sim y$ and $y \sim z$. Since $\ge$ itself is a transitive relation, we'll have $\text{age}(x) \ge \text{age}(z)$, giving $x \sim z$ and transitivity.
• Not Symmetric: If we have a strict inequality somewhere, there is no symmetry. For instance, if $X$ is $10$ years old and $Y$ is $15$ years old, then $Y \sim X$ but $X \not \sim Y$ (because $15 \ge 10$ but $10 < 15$).
• Not Antisymmetric: Suppose $x\sim y$ and $y \sim x$. The corresponding ages must be equal for this to hold, but this doesn't mean the people are the same person. For instance, both Paul and John can be ten years old, but they're obviously not the same person.