Certainly! Let's break down each of these logical statements involving the formula [tex]\( P(x) \)[/tex]:
### [tex]\( (\forall x) P(x) \)[/tex]:
This statement means "For all [tex]\( x \)[/tex], [tex]\( P(x) \)[/tex] is true."
- The symbol [tex]\(\forall\)[/tex] is the universal quantifier, which indicates that the statement [tex]\( P(x) \)[/tex] must hold for every possible value of [tex]\( x \)[/tex] within the domain we are considering.
### [tex]\( (\exists x) P(x) \)[/tex]:
This statement means "There exists an [tex]\( x \)[/tex] such that [tex]\( P(x) \)[/tex] is true."
- The symbol [tex]\(\exists\)[/tex] is the existential quantifier, which indicates that there is at least one value of [tex]\( x \)[/tex] in the domain for which [tex]\( P(x) \)[/tex] is true.
### [tex]\( (\forall x) \neg P(x) \)[/tex]:
This statement means "For all [tex]\( x \)[/tex], [tex]\( P(x) \)[/tex] is false."
- The universal quantifier [tex]\(\forall\)[/tex] combined with the negation [tex]\(\neg\)[/tex] indicates that for every possible value of [tex]\( x \)[/tex] in the domain, the statement [tex]\( P(x) \)[/tex] does not hold true.
### [tex]\( (\exists x) \neg P(x) \)[/tex]:
This statement means "There exists an [tex]\( x \)[/tex] such that [tex]\( P(x) \)[/tex] is false."
- The existential quantifier [tex]\(\exists\)[/tex] combined with the negation [tex]\(\neg\)[/tex] indicates that there is at least one value of [tex]\( x \)[/tex] in the domain for which [tex]\( P(x) \)[/tex] does not hold true.
### Summary of Concepts:
1. Universal Quantifier ([tex]\(\forall\)[/tex]): Implies that the statement must be true for every element in the domain.
2. Existential Quantifier ([tex]\(\exists\)[/tex]): Implies that there is at least one element in the domain for which the statement is true.
3. Negation ([tex]\(\neg\)[/tex]): Indicates the logical complement, i.e., the statement is not true.
### Interpretation in Natural Language:
- (a) [tex]\((\forall x) P(x)\)[/tex]: Every [tex]\( x \)[/tex] satisfies [tex]\( P(x) \)[/tex].
- (b) [tex]\((\exists x) P(x)\)[/tex]: There is some [tex]\( x \)[/tex] that satisfies [tex]\( P(x) \)[/tex].
- (c) [tex]\((\forall x) \neg P(x)\)[/tex]: Every [tex]\( x \)[/tex] does not satisfy [tex]\( P(x) \)[/tex].
- (d) [tex]\((\exists x) \neg P(x)\)[/tex]: There is some [tex]\( x \)[/tex] that does not satisfy [tex]\( P(x) \)[/tex].
These logical constructs are fundamental in mathematical logic and help us understand and formulate precise statements about the properties of elements within a given domain.
