Question

Let S be the Cartesian coordinate plane R×R and define a relation R on S by (a,b)R(c,d) iff a=c. Verify that R is an equivalence relation and describe a typical equivalence class E.

Answer 1

To verify that R is an equivalence relation, we need to show that it satisfies three properties: reflexive, symmetric, and transitive. A typical equivalence class E is a set of points that have the same x-coordinate.

Step-by-step explanation:

To verify that R is an equivalence relation, we need to show that it satisfies three properties: reflexive, symmetric, and transitive.

1. Reflexive: For any point (a,b) in S, (a,b) is related to itself since a=a. Therefore, R is reflexive.

2. Symmetric: If (a,b) is related to (c,d), then a=c. But since equality is symmetric, c=a. Therefore, (c,d) is related to (a,b), and R is symmetric.

3. Transitive: If (a,b) is related to (c,d) and (c,d) is related to (e,f), then a=c and c=e. But since equality is transitive, a=e. Therefore, (a,b) is related to (e,f), and R is transitive.

A typical equivalence class E is a set of points that have the same x-coordinate. For example, the equivalence class [2] would contain all points (2,b), where b is any real number.