Understanding Reflexive Relations

A relation R on a set A is defined as reflexive if for every element $a$ in set A, the pair $(a, a)$ is included in the relation R. Mathematically, this is expressed as:

$\forall a \in A, (a, a) \in R$.

Analyzing Intersection of Reflexive Relations ($R_1 \cap R_2$)

We are given that both $R_1$ and $R_2$ are reflexive relations on set A. This implies:

• For every $a \in A$, $(a, a) \in R_1$.
• For every $a \in A$, $(a, a) \in R_2$.

The intersection of two relations, $R_1 \cap R_2$, consists of all ordered pairs that are common to both $R_1$ and $R_2$. Since $(a, a)$ is present in $R_1$ and also in $R_2$ for all $a \in A$, it must be present in their intersection $R_1 \cap R_2$.

Therefore, $R_1 \cap R_2$ satisfies the condition $\forall a \in A, (a, a) \in R_1 \cap R_2$.

Conclusion: $R_1 \cap R_2$ is reflexive.

Analyzing Union of Reflexive Relations ($R_1 \cup R_2$)

Similar to the intersection, since $R_1$ and $R_2$ are reflexive:

• For every $a \in A$, $(a, a) \in R_1$.
• For every $a \in A$, $(a, a) \in R_2$.

The union of two relations, $R_1 \cup R_2$, consists of all ordered pairs that are present in $R_1$, or in $R_2$, or in both. Because $(a, a)$ is present in $R_1$ (and also in $R_2$) for all $a \in A$, it is guaranteed to be included in the union $R_1 \cup R_2$.

Therefore, $R_1 \cup R_2$ satisfies the condition $\forall a \in A, (a, a) \in R_1 \cup R_2$.

Conclusion: $R_1 \cup R_2$ is reflexive.

Final Result

Based on the analysis, both the intersection ($R_1 \cap R_2$) and the union ($R_1 \cup R_2$) of two reflexive relations $R_1$ and $R_2$ on a set A are themselves reflexive.