Final answer:
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.