I'm currently working on some first order logic questions as a brush up for a discrete mathematics course and I'm having a bit of trouble remembering exactly how to find the negation of a logical statment. Precisely, how do I rewrite certain statements as the negated versions of themselves? I've had luck with a couple but right now I'm stumped on the following:
At the moment I'm working on an assignment for a discrete mathematics course and I've run into a question that seems deceptively simple based on how it is asked but I feel like the instructor is asking for more. The question goes as follows:
Give the negation of the statement
∃! x ∈ U [P(x)].
But to negate this statement could I not just put a ¬ in front of the equation and be done? I only ask because on this assignment most questions are worth 4-6 marks and this particular question is worth 10 so I feel like there must be more to it. Any help is greatly appreciated.
In a trivial sense, yes you could just stick a $\neg$ at the beginning, but, similarly to saying that the solutions to $x^5+x^4+2x^2+3 = 0$ are those $x$ for which it is true, you probably aren't going to get any points.
So, the statement says, "there is a unique element of $U$ with property $P$". There are two ways in which this is false, either no element of $U$ has the property, or more than one does. We can express the first as $$ \forall x (x \in U \rightarrow (\neg P(X)))$$ and of course there are many other ways, and the second can be parsed as $$ \exists x \exists y (x\in U \land y \in U \land x \neq y \land P(x) \land P(y)) $$ so, one form of the statement we want is $$ (\forall x (x \in U \rightarrow (\neg P(X)))) \lor (\exists x \exists y (x\in U \land y \in U \land x \neq y \land P(x) \land P(y))). $$