Question

Suppose \( R_1 \) and \( R_2 \) are reflexive relations on a set \( A \). Which of the following statements is correct?

• A. \( R_1 \cap R_2 \) is reflexive and \( R_1 \cup R_2 \) is irreflexive
• B. \( R_1 \cap R_2 \) is irreflexive and \( R_1 \cup R_2 \) is reflexive
• C. Both \( R_1 \cap R_2 \) and \( R_1 \cup R_2 \) are irreflexive
• D. Both \( R_1 \cap R_2 \) and \( R_1 \cup R_2 \) are reflexive

Hint:

Remember that the intersection of reflexive relations is always reflexive, and the union of reflexive relations is also reflexive.

Answer 1

The Correct Option is D

Since \( R_1 \) and \( R_2 \) are reflexive relations, each element \( a \in A \) satisfies \( (a, a) \in R_1 \) and \( (a, a) \in R_2 \). 
The intersection \( R_1 \cap R_2 \) will also include \( (a, a) \) for every element \( a \in A \), making \( R_1 \cap R_2 \) reflexive. 
Similarly, the union \( R_1 \cup R_2 \) will include \( (a, a) \) for every \( a \in A \), making it reflexive as well.