Let R1 = \(a,b) | a b, a,b R\. Determine if R1 is reflexive, symmetric, antisymmetric, or transitive.

Step 1: Analyze the properties of the relation $R_1 = \{(a,b) | a \ge b, a,b \in R\}$.

• Reflexivity: A relation $R_1$ is reflexive if for every $a \in R$, $(a,a) \in R_1$. For any real number $a$, $a \ge a$ is true. Therefore, $(a,a) \in R_1$ for all $a \in R$. So, $R_1$ is reflexive.

• Symmetry: A relation $R_1$ is symmetric if for every $(a,b) \in R_1$, $(b,a) \in R_1