Question

Prove that the number of equivalence relations in the set \( \{1, 2, 3\} \) including \( \{(1, 2)\} \) and \( \{(2, 1)\} \) is 2.

Hint:

To prove equivalence relations, always verify reflexivity, symmetry, and transitivity properties.

Answer 1

Step 1: An equivalence relation on a set must be reflexive, symmetric, and transitive.

 Step 2: In the set \( \{1, 2, 3\} \), we start by considering the given relations \( \{(1, 2)\} \) and \( \{(2, 1)\} \). 

Step 3: Reflexive property requires that each element is related to itself, i.e., \( (1, 1), (2, 2), (3, 3) \) must be included. 

Step 4: Symmetric property requires that if \( (a, b) \) is in the relation, then \( (b, a) \) must also be in the relation. So, if \( (1, 2) \) exists, \( (2, 1) \) must also exist. 

Step 5: Transitive property ensures that if \( (a, b) \) and \( (b, c) \) are in the relation, then \( (a, c) \) must also be in the relation. 

Step 6: The equivalence relations on \( \{1, 2, 3\} \) that satisfy these conditions are: \[ R_1 = \{ (1, 1), (2, 2), (3, 3) \}, \quad R_2 = \{ (1, 2), (2, 1), (1, 1), (2, 2), (3, 3) \}. \] Thus, the number of equivalence relations is 2.