Consider the following properties: A. Reflexive B. Antisymmetric C. Symmetricβ¦
2020
Consider the following properties:
A. Reflexive
B. Antisymmetric
C. Symmetric
LetΒ π΄ = {π,π,π,π,π,π,π}Β andΒ π
={(π,π),(π,π),(π,π),(π,π),(π,π),(π,π),(π,π),(π,π)}Β be a relation onΒ π΄. Which of the following property (properties) is (are) satisfied by the relationΒ π
?
- A.
Only A
- B.
Only C
- C.
Both A and B
- D.
B and not A
Attempted by 398 students.
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Correct answer: D
Final answer: The relation satisfies only the antisymmetric property.
Reflexive: Not satisfied. A reflexive relation must contain (x,x) for every element x in A. Here (c,c) and (d,d) are missing, so reflexivity fails.
Symmetric: Not satisfied. A symmetric relation requires that whenever (x,y) is in R, (y,x) is also in R. For example, (c,d) is in R but (d,c) is not, so symmetry fails.
Antisymmetric: Satisfied. Antisymmetry means if (x,y) and (y,x) are both in R then x = y. There are no distinct elements x and y for which both (x,y) and (y,x) appear in R. The only mutual pairs present are of the form (x,x), which do not violate antisymmetry.
Conclusion: The relation is not reflexive and not symmetric, but it is antisymmetric. Therefore the correct choice is the one that states the relation is antisymmetric and not reflexive.
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