**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 $D$.

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**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 $Q$."

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**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.

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**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