Consider the following relations on {1,2,3,4}. Which of the following…
2024
Consider the following relations on {1,2,3,4}. Which of the following relations are reflexive?
(A) R1={(1,1),(1,2),(2,1),(2,2),(3,4),(4,1),(4,4)}
(B) R2={(1,1),(1,2),(2,1)}
(C) R3={(1,1),(1,2),(1,4),(2,1),(2,2),(3,3),(4,1),(4,4)}
(D)R4={(2,1),(3,1),(3,2),(4,1),(4,2),(4,3)}
(E) R5={(1,1),(1,2),(1,3),(1,4),(2,2),(2,3),(2,4),(3,3),(3,4),(4,4)}
Choose the correct answer from the options given below:
- A.
(A), (B) and (C) Only
- B.
(A), (D) and (E) Only
- C.
(D) and (E) Only
- D.
(C) and (E) Only.
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Correct answer: D
Definition: A relation on {1,2,3,4} is reflexive if it contains the diagonal pairs (1,1), (2,2), (3,3), and (4,4).
R1 = {(1,1),(1,2),(2,1),(2,2),(3,4),(4,1),(4,4)}: missing (3,3), so not reflexive.
R2 = {(1,1),(1,2),(2,1)}: missing (2,2), (3,3), and (4,4), so not reflexive.
R3 = {(1,1),(1,2),(1,4),(2,1),(2,2),(3,3),(4,1),(4,4)}: contains (1,1),(2,2),(3,3),(4,4), so reflexive.
R4 = {(2,1),(3,1),(3,2),(4,1),(4,2),(4,3)}: does not contain the diagonal pairs (1,1),(2,2),(3,3),(4,4), so not reflexive.
R5 = {(1,1),(1,2),(1,3),(1,4),(2,2),(2,3),(2,4),(3,3),(3,4),(4,4)}: contains (1,1),(2,2),(3,3),(4,4), so reflexive.
Conclusion: The reflexive relations are R3 and R5, so the correct answer is the option that lists R3 and R5 only.
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