If $R_1$ and $R_2$ are symmetric and transitive, prove that also $R_1 \cup R_2$ is symmetric and transitive

Question

I have to prove or confute (with a counter example) that, if $R_1$ and $R_2$ are symmetric and transitive, then $R_1 \cup R_2$ is also symmetric and transitive.

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Note: To prove the transitive case, I think I will ask a separate question, so that we can focus on the symmetric part now.

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The first thing I thought to do was to think about what does it mean $R_1$ and $R_2$ to be symmetric and transitive. Then I conclude that if a pair $(x, y) \in R_1$, then also $(y, x) \in R_1$.

I assumed also that $R_1$ and $R_2$ are define on a set $A = \{x, y, z \}$

I have to keep in my what is my goal: prove that $R_1 \cup R_2$ is symmetric, which means that if $((x, y) \in R_1 \cup R_2) \rightarrow ((y, x) \in R_1 \cup R_2)$. At this point, I could try to assume $(x, y) \in R_1 \cup R_2$ and prove $(y, x) \in R_1 \cup R_2$:

Assuming $(x, y) \in R_1 \cup R_2$, which means $(x, y) \in R_1 \lor (x, y) \in R_2$, I have to split my proof in 2 cases:

1. Case 1: I assume $(x, y) \in R_1$

   Since $R_1$ is symmetric, then $(y, x) \in R_1$. Since we know now that $(y, x) \in R_1$, we can say that $(y, x) \in R_1 \cup R_2$

2. Case 2: I assume $(x, y) \in R_2$

   Since $R_2$ is symmetric, then $(y, x) \in R_2$. Since we know now that $(y, x) \in R_2$, we can say that $(y, x) \in R_1 \cup R_2$

Is this a proof of the fact that, if $R_1$ and $R_2$ is symmetric, then also $R_1 \cup R_2$ is?

Answer 1

You might try this with a few examples of $R_1$ and $R_2$. For instance, consider the set of all line segments in the plane. If $A$ and $B$ are two line segments, let $A\mathrel{R_1}B$ if $A$ and $B$ have the same length, and $A \mathrel{R_2} B$ if $A$ and $B$ have the same slope (or are both vertical). Clearly $R_1$ and $R_2$ are symmetric and transitive (indeed, they are both equivalence relations). $R_1 \cup R_2$ is the relation that $A$ and $B$ have either the same length or the same slope (or both).

The conditions you need to check are:

• symmetry: If $A$ and $B$ have the same length or the same slope, do $B$ and $A$ have the same length or slope?
• transitivity: If $A$ and $B$ have the same length or the same slope, and $B$ and $C$ have the same length or the same slope, must $A$ and $C$ have the same length or the same slope? Draw some pictures and I think you might see the issue.

Symmetry looks pretty straightforward. IF $A$ has the same length as $B$ or $A$ is parallel to $B$, one of those has to be true. If $A$ has the same length as $B$, then $B$ has the same length as $A$. Or, if $A$ is parallel to $B$, then $B$ is parallel to $A$. In either case, the statement “$B$ has the same length as $A$ or $B$ is parallel to $A$” is true.

On the other hand...what if $A$ has the same length as $B$, but $B$ is parallel to $C$? Is it necessarily true that $A$ has the same length as $C$, or is parallel to $C$? It's not to hard to come up with a counterexample. For instance, $A$ and $B$ can be perpendicular but have the same length, and $C$ parallel to $B$ but twice as long.

This example ought to convince you that if $R_1$ and $R_2$ are symmetric, then $R_1 \cup R_2$ must be symmetric. However, if $R_1$ and $R_2$ are transitive, $R_1 \cup R_2$ need not be transitive.