Let π be the set of all people. Let π be a binary relation on π such thatβ¦
2019
LetΒ πΒ be the set of all people. LetΒ π Β be a binary relation onΒ πΒ such thatΒ (π,π)Β is inΒ π Β ifΒ πΒ is a brother ofΒ π. IsΒ π Β symmetric transitive, an equivalence relation, aΒ partial order relation?
- A.
NO, NO, NO, NO
- B.
NO, NO, YES, NO
- C.
NO, YES, NO, NO
- D.
NO, YES, YES, NO
Attempted by 276 students.
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Correct answer: A
Answer: The relation is not symmetric, not transitive, not an equivalence relation, and not a partial order.
Symmetric? No. If person A is a brother of person B, B might be female, so B is not a brother of A. Example: John is a brother of Mary, so (John, Mary) is in the relation, but (Mary, John) is not.
Transitive? No. Being a brother is not transitive: two brother links need not connect the endpoints. For example, let A, B, C be males with A and B sharing mother M1 (so A is brother of B) and B and C sharing father F1 (so B is brother of C). A and C may share no parent, so A is not a brother of C.
Equivalence relation? No. An equivalence relation must be reflexive, symmetric, and transitive. The relation fails reflexivity (no person is a brother of themselves) and fails symmetry and transitivity as shown above.
Partial order? No. A partial order requires reflexivity, antisymmetry, and transitivity. The relation is not reflexive, and antisymmetry fails (two distinct people can be brothers of each other), so it cannot be a partial order.
Therefore the correct classification is: not symmetric, not transitive, not an equivalence relation, and not a partial order (i.e., NO, NO, NO, NO).
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