To prove that if a relation $R$ on a set $A$ is transitive, then the inverse relation $R^{-1}$ is also transitive, we begin by recalling the definitions of transitivity and inverse relations.

Definitions:

1. Transitivity: A relation $R$ is transitive if whenever $(x, y) \in R$ and $(y, z) \in R$, it follows that $(x, z) \in R$.

   Symbolically, $R$ is transitive if: $(x, y) \in R \text{ and } (y, z) \in R \implies (x, z) \in R.$

2. Inverse relation: For a relation $R$, the inverse relation $R^{-1}$ is defined as: $R^{-1} = \{(y, x) : (x, y) \in R\}.$

We aim to show that if $R$ is transitive, then $R^{-1}$ is also transitive.

Proof:

Let $R$ be a transitive relation on a set $A$.

Consider the inverse relation $R^{-1}$. To show that $R^{-1}$ is transitive, we need to prove the following:
For any $a, b, c \in A$, $(a, b) \in R^{-1} \text{ and } (b, c) \in R^{-1} \implies (a, c) \in R^{-1}.$

Step 1: Assume $(a, b) \in R^{-1}$ and $(b, c) \in R^{-1}$.

By the definition of inverse relations:

• $(a, b) \in R^{-1}$ means $(b, a) \in R$.
• $(b, c) \in R^{-1}$ means $(c, b) \in R$.

Step 2: Use the transitivity of $R$.

Since $(b, a) \in R$ and $(c, b) \in R$, and $R$ is transitive, it follows that: $(c, a) \in R.$

Step 3: Apply the definition of inverse relations again.

Since $(c, a) \in R$, by the definition of $R^{-1}$, this implies: $(a, c) \in R^{-1}.$

Conclusion:

We have shown that if $(a, b) \in R^{-1}$ and $(b, c) \in R^{-1}$, then $(a, c) \in R^{-1}$. Therefore, $R^{-1}$ is transitive.

Thus, if $R$ is transitive, then its inverse relation $R^{-1}$ is also transitive, as required.

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Would you like further details or clarification? Here are five questions to explore this topic further:

1. Can a relation be transitive but its inverse not be transitive?
2. What are some examples of transitive and non-transitive relations?
3. How does symmetry of a relation relate to its transitivity?
4. If $R^{-1}$ is transitive, is $R$ necessarily transitive?
5. Can the composition of a relation with its inverse lead to a transitive relation?

Tip: Understanding the difference between relations and functions helps in visualizing how properties like transitivity apply to both.