Suppose that the domain of discourse is the set {-2, -1, 0, 1, 2}. Express the following statements without using quantifiers, using only the connectives ¬, V, Λ.
(a) ∃x P(x)
(b) ∀x P(x)
(c) ∃ ¬P(x)
(d) ∀ ¬P(x)
(e) ¬∃ P(x)
(f) ¬∀ P(x)
To express the given statements without using quantifiers, we can use the connectives ¬ (negation), V (disjunction), and Λ (conjunction). Here are the expressions for each statement:
(a) ∃x P(x): There exists an x such that P(x).
This can be expressed as P(-2) V P(-1) V P(0) V P(1) V P(2).
(b) ∀x P(x): For all x, P(x).
This can be expressed as P(-2) Λ P(-1) Λ P(0) Λ P(1) Λ P(2).
(c) ∃ ¬P(x): There exists an x such that ¬P(x).
This can be expressed as ¬P(-2) V ¬P(-1) V ¬P(0) V ¬P(1) V ¬P(2).
(d) ∀ ¬P(x): For all x, ¬P(x).
This can be expressed as ¬P(-2) Λ ¬P(-1) Λ ¬P(0) Λ ¬P(1) Λ ¬P(2).
(e) ¬∃ P(x): It is not true that there exists an x such that P(x).
This can be expressed as ¬(P(-2) V P(-1) V P(0) V P(1) V P(2)).
(f) ¬∀ P(x): It is not true that for all x, P(x).
This can be expressed as ¬(P(-2) Λ P(-1) Λ P(0) Λ P(1) Λ P(2)).
These expressions convey the same meaning as the original statements and use only the given connectives.