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 using only the connectives ¬, V, Λ, we can use the following equivalences:
¬(∃x P(x)) ≡ ∀x ¬P(x) -- ¬(there exists) is equivalent to for all not
¬(∀x P(x)) ≡ ∃x ¬P(x) -- ¬(for all) is equivalent to there exists not
Now, let's express each statement without using quantifiers:
(a) ∃x P(x)
This statement asserts that there exists at least one element x in the domain for which P(x) is true. Using the equivalence mentioned above, we can rewrite it as:
¬(∀x ¬P(x))
(b) ∀x P(x)
This statement claims that for every element x in the domain, P(x) is true. No negation is required.
(c) ∃ ¬P(x)
This statement asserts that there exists at least one element x in the domain for which P(x) is false. Using the equivalence mentioned above, we can rewrite it as:
¬(∀x P(x))
(d) ∀ ¬P(x)
This statement claims that for every element x in the domain, P(x) is false. No negation is required.
(e) ¬∃ P(x)
This statement asserts that it is not the case that there exists at least one element x in the domain for which P(x) is true. Using the equivalence mentioned above, we can rewrite it as:
∀x ¬P(x)
(f) ¬∀ P(x)
This statement asserts that it is not the case that for every element x in the domain, P(x) is true. Using the equivalence mentioned above, we can rewrite it as:
∃x ¬P(x)
By using these equivalences, we have expressed each statement without using quantifiers and using only the connectives ¬, V, Λ.