149 lines
4.8 KiB
Markdown
149 lines
4.8 KiB
Markdown
**Test Yourself**
|
|
|
|
Page 141
|
|
|
|
1. If $P(x)$ is a predicate with domain $D$, the truth set of $P(x)$ is denoted
|
|
_______. We read these symbols out loud as _______.
|
|
|
|
$\{x \in D | P(x)\}$; "the set of all $x$ in $D$ such that $P(x)$."
|
|
|
|
2. Some ways to express the symbol $\forall$ in words are _______.
|
|
|
|
for every for all, for any, for each, for arbitrary, given any
|
|
|
|
3. Some ways to express the symbol $\exists$ in words are _______.
|
|
|
|
there exists, there exist, there exists at least one, for some, for at least
|
|
one, we can find a
|
|
|
|
4. A statement of the form $\forall x \in D$, $Q(x)$ is true if, and only if,
|
|
$Q(x)$ is _______ for _______.
|
|
|
|
true; every $x$ in $D$.
|
|
|
|
5. A statement of the form $\exists x \in D$ such that $Q(x)$ is true if, and
|
|
only if, $Q(x)$ is _______ for _______.
|
|
|
|
true; at least one $x$ in $D$.
|
|
|
|
---
|
|
|
|
**Test Yourself**
|
|
|
|
Page 152
|
|
|
|
1. A negation for "All $R$ have property $S$" is "There is _______ $R$ that
|
|
_______."
|
|
|
|
exists at least one; does not have property $S$.
|
|
|
|
2. A negation for "Some $R$ have property $S$" is "_______."
|
|
|
|
"No $R$ have property $S$."
|
|
|
|
3. A negation for "For every $x$, if $x$ has property $P$ then $x$ has property
|
|
$Q$" is "_______."
|
|
|
|
"There exists at least one $x$ such that $x$ has property $P$ and $x$ does not
|
|
have property $Q$."
|
|
|
|
4. The converse of "For every $x$, if $x$ has property $P$ then $x$ has property
|
|
$Q$" is "_______."
|
|
|
|
"For every $x$, if $x$ has property $Q$ then $x$ has property $P$."
|
|
|
|
5. The contrapositive of "For every $x$, if $x$ has property $P$ then $x$ has
|
|
property $Q$" is "_______."
|
|
|
|
"For every $x$, if $x$ does not have property $Q$, then $x$ does not have
|
|
property $P$."
|
|
|
|
6. The inverse of "For every $x$, if $x$ has property $P$ then $x$ has property
|
|
$Q$" is "_______."
|
|
|
|
"For every $x$, if $x$ does not have property $P$, then $x$ does not have
|
|
property $Q$."
|
|
|
|
---
|
|
|
|
**Test Yourself**
|
|
|
|
Page 165
|
|
|
|
1. To establish the truth of a statement of the form
|
|
"$\forall x \text{ in } D, \exists y \text{ in } E \text{ such that } P(x)$,"
|
|
you imagine that someone has given you an element $x$ from $D$ but that you
|
|
have no control over what that element is. Then you need to find _______ with
|
|
the property that the $x$ the person gave you together with the _______ you
|
|
subsequently found satisfy _______.
|
|
|
|
$y \in E$; $y$; $P(x, y)$
|
|
|
|
2. To establish the truth of a statement of the form
|
|
"$\exists x \text{ in } D \text{ such that } \forall y \text{ in } E, P(x, y)$,"
|
|
you need to find _______ so that no matter what _______ a person might
|
|
subsequently give you, _______ will be true.
|
|
|
|
$x \in D$; $y \in E$; $P(x, y)$
|
|
|
|
3. Consider the statement
|
|
"$\forall x, \exists y \text{ such that } P(x, y), \text{ a property involving } x \text{ and } y, \text{ is true}$."
|
|
A negation for this statement is "_______."
|
|
|
|
"$\exists x$ such that $\forall y$, the property $P(x, y)$ is false."
|
|
|
|
4. Consider the statement
|
|
"$\exists x \text{ such that } \forall y, P(x, y), \text{ a property involving } x \text{ and } y, \text{ is true}$."
|
|
A negation for this statement is "_______."
|
|
|
|
"$\forall x, \exists y$ such that the property $P(x, y)$ is false."
|
|
|
|
5. Suppose $P(x, y)$ is some property involving $x$ and $y$, and suppose the
|
|
statement
|
|
"$\forall x \text{ in } D, \exists y \text{ in } E \text{ such that } P(x, y)$"
|
|
is true. Then the statement
|
|
"$\exists x \text{ in } D \text{ such that } \forall y \text{ in } E, P(x, y)$"
|
|
|
|
a. is true.
|
|
|
|
b. is false.
|
|
|
|
c. may be true or may be false.
|
|
|
|
c is the answer, it may be true or false depending on the nature of the property
|
|
involving $x$, and $y$. In other words, it relies on what $P(x, y)$ states.
|
|
|
|
---
|
|
|
|
**Test Yourself**
|
|
|
|
Page 179
|
|
|
|
1. The rule of universal instantiation says that if some property is true for
|
|
_______ in a domain, then it is true for _______.
|
|
|
|
all elements; any particular element in the domain
|
|
|
|
2. If the first two premises of universal modus ponens are written as "If $x$
|
|
makes $P(x)$ true, then $x$ makes $Q(x)$ true" and "For a particular value of
|
|
$a$ _______ , " then the conclusion can be written as "______. "
|
|
|
|
$P(a)$ is true, $Q(a)$ is true
|
|
|
|
3. If the first two premises of universal modus tollens are written as "If $x$
|
|
makes $P(x)$ true, then $x$ makes $Q(x)$ true" and "For a particular value of
|
|
$a$ _______ ," then the conclusion can be written as " _______. "
|
|
|
|
$Q(a)$ is false; $P(a)$ is false
|
|
|
|
4. If the first two premises of universal transitivity are written as "Any $x$
|
|
that makes $P(x)$ true makes $Q(x)$ true" and "Any $x$ that makes $Q(x)$ true
|
|
makes $R(x)$ true," then the conclusion can be written as "_______."
|
|
|
|
"Any $x$ that makes $P(x)$ true makes $R(x)$ true"
|
|
|
|
5. Diagrams can be helpful in testing an argument for validity. However, if some
|
|
possible configurations of the premises are not drawn, a person could
|
|
conclude that an argument was _______ when it was actually _______.
|
|
|
|
valid; invalid
|