Let L be a set with a relation R that is reflexive, antisymmetric and…
1999
Let L be a set with a relation R that is reflexive, antisymmetric and transitive. Also, for every pair of elements a, b ∈ L, the least upper bound lub(a, b) and greatest lower bound glb(a, b) exist. Which of the following are true?
a) L is a poset.
b) L is a Boolean algebra.
c) L is a lattice.
d) None of the above.
- A.
a, b
- B.
a, c
- C.
only c
- D.
only b
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Correct answer: B
The correct answer is: a, c.
A relation that is reflexive, antisymmetric and transitive is a partial order. Therefore (L, R) is a partially ordered set, i.e. a poset. So statement a is true.
A lattice is a poset in which every pair of elements has both:
1. a least upper bound (join), and
2. a greatest lower bound (meet).
The question states that lub(a, b) and glb(a, b) exist for every pair a, b ∈ L. Therefore L is a lattice, so statement c is true.
A Boolean algebra needs additional properties such as boundedness, complements and distributivity. These are not guaranteed here. So statement b is not necessarily true.