Question

Does the relation defined by \( R = \{(x, y) : y \text{ is divisible by } x\} \) on the set \( A = \{1, 2, 3, 4, 5, 6\} \), an equivalence relation?

Hint:

To disprove a property like symmetry or transitivity, you only need to find one counterexample. To prove it, you must show it holds for all possible cases. Forgetting a single property check can lead to an incorrect conclusion.

Answer 1

Step 1: Understanding the Concept:
For a relation to be an equivalence relation, it must satisfy three properties: reflexivity, symmetry, and transitivity. We need to test the given relation R for each of these properties.
Step 2: Key Definitions:
- Reflexive: For all \( a \in A \), \( (a, a) \in R \). (Is every element related to itself?)
- Symmetric: If \( (a, b) \in R \), then \( (b, a) \in R \). (If a is related to b, is b related to a?)
- Transitive: If \( (a, b) \in R \) and \( (b, c) \in R \), then \( (a, c) \in R \). (If a is related to b and b is related to c, is a related to c?)
Step 3: Detailed Explanation:
1. Test for Reflexivity:
For any element \( x \in A \), the pair \( (x, x) \) is in R if 'x is divisible by x'. Every number is divisible by itself. Therefore, \( (x, x) \in R \) for all \( x \in A \). The relation is reflexive.
2. Test for Symmetry:
We need to check if \( (x, y) \in R \) implies \( (y, x) \in R \). This means, if 'y is divisible by x', does it imply that 'x is divisible by y'?
Let's take a counterexample from the set A.
Consider the pair (2, 4). Since 4 is divisible by 2, \( (2, 4) \in R \).
Now, we must check if the reverse is true, i.e., if \( (4, 2) \in R \). This would mean that 2 is divisible by 4. This is false.
Since we found a counterexample, the relation is not symmetric.
3. Conclusion:
An equivalence relation must be reflexive, symmetric, and transitive. Since the relation R is not symmetric, it cannot be an equivalence relation. We do not need to check for transitivity.
Step 4: Final Answer:
No, the given relation is not an equivalence relation because it is not symmetric.