Suppose that R1 and R2 are reflexive relations on a set A. Which of the…
2016
Suppose that R1 and R2 are reflexive relations on a set A.
Which of the following statements is correct ?
- A.
R1 ∩ R2 is reflexive and R1 ∪ R2 is irreflexive.
- B.
R1 ∩ R2 is irreflexive and R1 ∪ R2 is reflexive.
- C.
Both R1 ∩ R2 and R1 ∪ R2 are reflexive.
- D.
Both R1 ∩ R2 and R1 ∪ R2 are irreflexive.
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Correct answer: C
Answer: Both R1 ∩ R2 and R1 ∪ R2 are reflexive.
Definition: A relation R on a set A is reflexive if for every element a in A, the pair (a,a) belongs to R.
Intersection (R1 ∩ R2): For any a in A, reflexivity of R1 and R2 gives (a,a) ∈ R1 and (a,a) ∈ R2. Therefore (a,a) ∈ R1 ∩ R2 for every a, so R1 ∩ R2 is reflexive.
Union (R1 ∪ R2): For any a in A, since (a,a) ∈ R1 and (a,a) ∈ R2, we have (a,a) ∈ R1 ∪ R2. Thus R1 ∪ R2 is reflexive.
Note: Any statement asserting that either the intersection or the union is irreflexive is false here, because irreflexive relations contain no diagonal pairs (a,a), which contradicts reflexivity of the original relations.
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