Suppose that the domain of the propositional function $P(x)$ consists of the integers -2, -1, 0, 1 and 2. Write out each of these propositions using disjunctions, conjunctions and negations.
a) $\exists x \ P(x)$ So it says that there exists an $x$ for which $P(x)$ is true.
Then $P(-2) \vee P(-1) \vee P(0) \vee P(1) \vee P(2)$
b) $\forall x \ P(x)$ Here it says that $P(x)$ is true for every $x$.
Then $P(-2) \wedge P(-1) \wedge P(0) \wedge P(1) \wedge P(2)$
c) $\exists x \ \neg P(x)$
Am I right in a) and b). But I am not sure about c)
$\endgroup$1 Answer
$\begingroup$A and B are correct. C can be read as "There is at least one x for which P(x) is not true." This is the inverse of part B, which says "P(x) is true for all x."
Therefore part C is
$\neg(P(-2) \wedge P(-1) \wedge P(0) \wedge P(1) \wedge P(2))$
This can also be written as
$\neg P(-2) \vee \neg P(-1) \vee \neg P(0) \vee \neg P(1) \vee \neg P(2) \vee$
These two statements are equivalent, and are both correct.
$\endgroup$ 1
