As mentioned in the text, the notation ∃! x P(x) denotes "There exists a unique x such that P(x)

step 1 We know with the notation, there exists a unique x, p of x, means just that. And so we want to f

As mentioned in the text, the notation ∃! x P(x) denotes...

As mentioned in the text, the notation $\exists ! x P(x)$ denotes "There exists a unique $x$ such that $P(x)$ is true." If the domain consists of all integers, what are the truth values of these statements? $$ \begin{array}{ll}{\text { a) } \exists ! x(x>1)} & {\text { b) } \exists ! x\left(x^{2}=1\right)} \\ {\text { c) } \quad \exists ! x(x+3=2 x)} & {\text { d) } \exists ! x(x=x+1)}\end{array} $$

As mentioned in the text, the notation $\exists ! x P(x)$ denotes
"There exists a unique $x$ such that $P(x)$ is true."
If the domain consists of all integers, what are the truth values of these statements?
$$
\begin{array}{ll}{\text { a) } \exists ! x(x>1)} & {\text { b) } \exists ! x\left(x^{2}=1\right)} \\ {\text { c) } \quad \exists ! x(x+3=2 x)} & {\text { d) } \exists ! x(x=x+1)}\end{array}
$$

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Key Concepts

Predicate Logic Evaluation

Predicate logic involves assessing statements formulated with quantifiers and predicates over a given domain. Evaluating such expressions requires checking if there exists exactly one integer that satisfies the predicate, whether there are multiple integers or none.

Equation Solving in Integers

When solving equations within the integer domain, one must consider that the solutions are confined to integers. Some equations might have no solution, exactly one solution, or multiple solutions within this restricted set, directly affecting the truth of uniqueness statements.

Uniqueness Quantifier

The uniqueness quantifier (?!) asserts that there is exactly one element in the domain that satisfies a given property. It is a combination of both the existence condition (at least one element) and the uniqueness condition (no more than one element) which must be met concurrently.

Domain of Discourse

The domain of discourse determines the set of all elements over which the logical expressions and quantifiers are evaluated. In this case, the domain is the set of all integers, which restricts the evaluation of the properties to integer values only.

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