60 lines
1.7 KiB
Markdown
60 lines
1.7 KiB
Markdown
Page 132
|
|
|
|
**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.
|
|
|
|
---
|
|
|
|
Page 132
|
|
|
|
**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
|
|
|
|
$$ \{x \in D | P(x)\} $$
|
|
|
|
---
|
|
|
|
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.
|
|
|
|
---
|
|
|
|
Page 134
|
|
|
|
**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 $D$.
|
|
|
|
---
|
|
|
|
Page 140
|
|
|
|
**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,
|
|
$\forall x, P(x) \leftrightarrow Q(x)$.
|