Relations

Step 1: Understand the relation.
The relation defines pairs where . Hence, we list all pairs satisfying this condition.
Step 2: Conclusion.
Thus, the relation is .
Final Answer:

We are given the relation defined on the set of natural numbers as: This means that or , i.e., the difference between and is exactly 3. Step 1: Understand the condition for the relation. The relation pairs numbers where the difference between them is exactly 3. So, for example: - If , then (since ). - If , then (since ). - If , then (since ). Step 2: Match the given options. From the given options: - Option (a): shows pairs where is smaller than by 3, but the problem specifies that is larger than , so this is incorrect. - Option (b): correctly reflects that differs from by 3 in the right order (since , , and ). - Option (c): does not satisfy the condition that differs from by exactly 3. Thus, the correct answer is (b).

Step 1: Verifying equivalence properties.
The relation satisfies the reflexive, symmetric, and transitive properties. Therefore, it is an equivalence relation.

Step 2: Conclusion.
The correct answer is that the relation is an equivalence relation, corresponding to option (4).

Step 1: Identifying the relation.
The relation is defined as , so each element of the set corresponds to the absolute value of . The correct set corresponds to option (1).

Step 2: Conclusion.
The correct answer is option (1).