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**Definition**

A **predicate** is a sentence that contains a finite number of variables and
becomes a statement when specific values are substituted for the variables. The
**domain** of a predicate variable is the set of all values that may be
substituted in place of the variable.

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**Definition**

If $P(x)$ is a predicate and $x$ has domain $D$, the **truth set** of $P(x)$ is
the set of all elements of $D$ that make $P(x)$ true when they are substituted
for $x$. The truth set of $P(x)$ is denoted

$$ \{x \in D | P(x)\} $$

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Page 133

**Definition**

Let $Q(x)$ be a predicate and $D$ the domain of $x$. A **universal statement**
is a statement of the form "$\forall x \in D, Q(x)$." It is defined to be true
if, and only if, $Q(x)$ is true for each individual $x$ in $D$. It is defined to
be false if, and only if, $Q(x)$ is false for at least one $x$ in $D$. A value
for $x$ for which $Q(x)$ is false is called a **counterexample** to the
universal statement.

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**Definition**

Let $Q(x)$ be a predicate and $D$ the domain of $x$. An **existential
statement** is a statement of the form "$\exists x \in D$ such that $Q(x)$." It
is defined to be true if, and only if, $Q(x)$ is true for at least one $x$ in
$D$. It is false if, and only if, $Q(x)$ is false for all $x$ in $D$.

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**Notation**

Let $P(x)$ and $Q(x)$ be predicates and suppose the domain of $x$ is $D$.

- The notation $P(x) \Rightarrow Q(x)$ means that every element in the truth set
  of $P(x)$ is in the truth set of $Q(x)$, or, equivalently,
  $\forall x, P(x) \to Q(x)$.

- The notation $P(x) \Leftrightarrow Q(x)$ means that $P(x)$ and $Q(x)$ have
  identical truth sets, or, equivalently,
  $\forall x, P(x) \leftrightarrow Q(x)$.