If `R` and `S` are transitive relations on a set `A` , then prove that `RuuS` may not be a transitive relation on `A` .

Explanation

To prove that the union of two transitive relations \( R \) and \( S \) on a set \( A \) may not be transitive, we can follow these steps:

### Step 1: Define the Set and Relations
Let \( A = \{ A, B, C \} \). Define two transitive relations \( R \) and \( S \) on set \( A \).

- Let \( R = \{ (A, A), (A, B), (B, A), (B, B) \} \)
- Let \( S = \{ (B, B), (B, C), (C, C) \} \)

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