Multiple choice

Let (S,$\le$) be a partial order with two minimal elements a and b, and a maximum element c. Let P: S {True, False} be a predicate defined on S. Suppose that P(a) = True, P(b) = False and P(x)$\Rightarrow$P(y) for all x, y $\epsilon$S satisfying x$\le$y,

Where $\Rightarrow$stands for logical implication. Which of the following statements CANNOT be true?

  1. P(x) = True for all x $\epsilon$S such that x $\ne$b
  2. P(x) = False for all x $\epsilon$S such that x $\ne$a and x $\ne$c
  3. P(x) = False for all x $\epsilon$S such that b $\le$x and x $\ne$c
  4. P(x) = False for all x $\epsilon$S such that a $\le$x and b $\le$x
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

rder=0>P(y) If P(x) then P(y) and a, y $\epsilon$S, satisfies x $\le$ y So, -P (a) = True, p(b) = false $\Rightarrow$False $\cup$False $\Rightarrow$False So, P(x) = False for all x $\epsilon$S, such that b $\le$ x and x $\ne$c.

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