Let (A, ≤) be a partial order with two minimal elements a, b and a maximum element c. Let P:A –> {True, False} be a predicate defined on A. Suppose that P(a) = True, P(b) = False and P(a) ⇒ P(b) for all satisfying a ≤ b, where ⇒ stands for logical implication. Which of the following statements cannot be true?
A. P(x) = True for all x S such that x ≠ b
B. P(x) = False for all x ∈ S such that b ≤ x and x ≠ c
C. P(x) = False for all x ∈ S such that x ≠ a and x ≠ c
D. P(x) = False for all x ∈ S such that a ≤ x and b ≤ x
Answer:D
Let (A, ≤) be a partial order with two minimal elements a, b and a maximum element c. Let P:A –> {True, False} be a predicate defined on A. Suppose that P(a) = True, P(b) = False and P(a) ⇒ P(b) for all satisfying a ≤ b, where ⇒ stands for logical implication. Which of the following statements cannot be true?
A. P(x) = True for all x S such that x ≠ b B. P(x) = False for all x ∈ S such that b ≤ x and x ≠ c C. P(x) = False for all x ∈ S such that x ≠ a and x ≠ c D. P(x) = False for all x ∈ S such that a ≤ x and b ≤ x
Explanation: Here, maximum element is c and so c is of a higher order than any other element in A. Minimal elements are a and b: No other element in A is of lower order than either a or b.