Question

Examine that the relation \( R \) in the set \( \{1, 2, 3, 4\} \) given by \( R = \{(1, 2), (2, 2), (1, 1), (4, 4), (1, 3), (3, 3), (3, 2)\} \) is reflexive and transitive but not symmetric.

Hint:

To check reflexivity, ensure that each element of the set is related to itself. To check transitivity, verify that for any two pairs \( (a, b) \) and \( (b, c) \), the pair \( (a, c) \) also exists. Symmetry requires that if \( (a, b) \) is in the relation, then \( (b, a) \) must also be present.

Answer 1

Step 1: Checking reflexivity.
A relation is reflexive if for every element \( x \) in the set, the pair \( (x, x) \) belongs to the relation. In this case, the set is \( \{1, 2, 3, 4\} \). The pairs \( (1, 1), (2, 2), (3, 3), (4, 4) \) are all present in the relation, so the relation is reflexive.

Step 2: Checking transitivity.
A relation is transitive if whenever \( (a, b) \) and \( (b, c) \) are in the relation, then \( (a, c) \) must also be in the relation. - We have \( (1, 2) \) and \( (2, 3) \), so \( (1, 3) \) must also be present, which it is. - We also have \( (3, 2) \) and \( (2, 2) \), so \( (3, 2) \) is present. Since all necessary pairs are present, the relation is transitive.

Step 3: Checking symmetry.
A relation is symmetric if whenever \( (a, b) \) is in the relation, \( (b, a) \) must also be in the relation. In this case, \( (1, 2) \) is in the relation, but \( (2, 1) \) is not, so the relation is not symmetric.

Step 4: Conclusion.
The relation is reflexive and transitive, but not symmetric.