Let `R` and `S` be two non-void relations on a set A. Which of the following statements is false?

Options

A. R and S are transitive `implies R uu S` is transitive
B. R and S are transitive implies `R nn S` is symmetric
C. R and S are symmetric implies `R uu S` is symmetric
D. R and S are reflexive implies `R nn S` is reflexive

Explanation

To determine which statement is false regarding the relations \( R \) and \( S \) on a set \( A \), we will analyze each statement one by one.

### Step 1: Understand the Definitions
- **Transitive Relation**: A relation \( R \) is transitive if for all \( a, b, c \in A \), if \( (a, b) \in R \) and \( (b, c) \in R \), then \( (a, c) \in R \).
- **Symmetric Relation**: A relation \( R \) is symmetric if for all \( a, b \in A \), if \( (a, b) \in R \), then \( (b, a) \in R \).
- **Reflexive Relation**: A relation \( R \) is reflexive if for all \( a \in A \), \( (a, a) \in R \).

### Step 2: Analyze Each Statement
1. **Statement 1**: \( R \) and \( S \) are transitive implies \( R \cup S \) is transitive.
- **Analysis**: This statement is generally true. If both \( R \) and \( S \) are transitive, their union \( R \cup S \) may not necessarily be transitive.

2. **Statement 2**: \( R \) and \( S \) are transitive implies \( R \cap S \) is symmetric.
- **Analysis**: This statement is false. The intersection of two transitive relations is not guaranteed to be symmetric.

3. **Statement 3**: \( R \) and \( S \) are symmetric implies \( R \cup S \) is symmetric.
- **Analysis**: This statement is true. If both relations are symmetric, their union will also be symmetric.

4. **Statement 4**: \( R \) and \( S \) are reflexive implies \( R \cap S \) is reflexive.
- **Analysis**: This statement is true. If both relations are reflexive, their intersection will also be reflexive.

### Conclusion
The false statement among the given options is **Statement 2**: \( R \) and \( S \) are transitive implies \( R \cap S \) is symmetric.