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)
I have seen your post several times. Have not answered it since I am not familiar with the topic.
The last time I would have seen something like that must be about 55 years ago and it has long since gone into my "delete" part of my memory.
As I recall,
(a) ∃x P(x)
means "there exists an x such that P)x) is true. So, in this case, that wold be
P(-2) V P(-1) V P(0) V P(1) V P(2)
Hmmm. Not sure what (c) ∃ ¬P(x) means. There exists "not P(x)"?
Is that the same as
∃x ¬P(x), that is
there exist x such that P(x) is false
??
In any case, it appears that you just have to replace P(x) with suitable connections of P(-2)...P(2)
To express the given statements without using quantifiers, we can use logical connectives ¬ (negation), V (disjunction), and Λ (conjunction). Here's how we can express each statement:
(a) ∃x P(x): There exists an x such that P(x).
In this case, we can express this statement as P(-2) V P(-1) V P(0) V P(1) V P(2).
(b) ∀x P(x): For all x, P(x) is true.
We can express this statement 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) is true.
This statement 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) is true.
We can express this statement 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 statement 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) is true.
This statement can be expressed as ¬(P(-2) Λ P(-1) Λ P(0) Λ P(1) Λ P(2)).
By using the connectives ¬, V, and Λ, these statements can be expressed without using quantifiers.