Question

29. Let A = R × R. A relation S is defined on A as follows: For

all (x1, y1) and (x2, y2) in A, (x1, y1) S (x2, y2) ⇔ y1 = y2 In
9–33 determine whether the given relation is reflexive, symmetric,

Answer 1

The given relation S is reflexive but not symmetric.

Step-by-step explanation:

The given relation S is defined on A = R × R as follows: For all (x1, y1) and (x2, y2) in A, (x1, y1) S (x2, y2) ⇔ y1 = y2. We need to determine whether the given relation S is reflexive, symmetric, or transitive.

To check if the relation is reflexive, we need to verify if every element (x, y) in A is related to itself. In this case, if y1 = y1, which is true for every y, then the relation is reflexive.

To check if the relation is symmetric, we need to verify if whenever (x1, y1) is related to (x2, y2), then (x2, y2) is related to (x1, y1). Since the relation (x1, y1) S (x2, y2) is based on the equality of y1 and y2, it is not true that y1 = y2 implies y2 = y1. Hence, the relation is not symmetric.