If R and S be two non-void relations on a set A, which of the following statements if false?

Options

A. R and S are transitive `rarrRuuS` is transitive
B. R and S are transitive `rarrRnnS` is transitive
C. R and S are symmetric `rarrRuuS` is symmetric
D. None of these

Answer

To determine which statement is false regarding the relations R and S on the set A, we will analyze the properties of these relations step by step.

### Step 1: Define the Set and Relations
Let the set \( A = \{ A, B, C \} \).

We define the relations as follows:
- Relation \( R = \{ (A, A), (B, B), (A, B) \} \)
- Relation \( S = \{ (B, B), (C, C), (B, C) \} \)

### Step 2: Check Transitivity of R and S
A relation is transitive if whenever \( (x, y) \) and \( (y, z) \) are in the relation, then \( (x, z) \) must also be in the relation.

**For Relation R:**
- We have \( (A, B) \) and \( (B, B) \) (which is in R), but we do not have \( (A, B) \) leading to any new pairs.
- There are no other combinations that violate transitivity.
- Thus, \( R \) is transitive.

**For Relation S:**
- We have \( (B, C) \) and \( (C, C) \), but we do not have \( (B, C) \) leading to any new pairs.
- There are no other combinations that violate transitivity.
- Thus, \( S \) is also transitive.

### Step 3: Find R Union S
Now, we find the union of the two relations \( R \) and \( S \):
\[ R \cup S = \{ (A, A), (B, B), (A, B), (C, C), (B, C) \} \]

### Step 4: Check Transitivity of R Union S
To check if \( R \cup S \) is transitive:
- We see \( (A, B) \) and \( (B, C) \) are in \( R \cup S \), but \( (A, C) \) is not in \( R \cup S \).
- Therefore, \( R \cup S \) is **not transitive**.

### Step 5: Find R Intersection S
Next, we find the intersection of the two relations \( R \) and \( S \):
\[ R \cap S = \{ (B, B) \} \]

### Step 6: Check Transitivity of R Intersection S
The intersection \( R \cap S \) contains only one pair \( (B, B) \), which is trivially transitive since there are no other elements to violate transitivity.

### Conclusion
From our analysis:
- \( R \) is transitive.
- \( S \) is transitive.
- \( R \cup S \) is **not transitive**.
- \( R \cap S \) is transitive.

Thus, the false statement is related to the transitivity of \( R \cup S \).

### Final Answer
The false statement is that \( R \cup S \) is transitive.

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