114 lines
3.6 KiB
Markdown
114 lines
3.6 KiB
Markdown
**Test Yourself**
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Page 141
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1. If $P(x)$ is a predicate with domain $D$, the truth set of $P(x)$ is denoted
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_______. We read these symbols out loud as _______.
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$\{x \in D | P(x)\}$; "the set of all $x$ in $D$ such that $P(x)$."
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2. Some ways to express the symbol $\forall$ in words are _______.
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for every for all, for any, for each, for arbitrary, given any
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3. Some ways to express the symbol $\exists$ in words are _______.
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there exists, there exist, there exists at least one, for some, for at least
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one, we can find a
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4. A statement of the form $\forall x \in D$, $Q(x)$ is true if, and only if,
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$Q(x)$ is _______ for _______.
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true; every $x$ in $D$.
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5. A statement of the form $\exists x \in D$ such that $Q(x)$ is true if, and
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only if, $Q(x)$ is _______ for _______.
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true; at least one $x$ in $D$.
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---
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**Test Yourself**
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Page 152
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1. A negation for "All $R$ have property $S$" is "There is _______ $R$ that
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_______."
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exists at least one; does not have property $S$.
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2. A negation for "Some $R$ have property $S$" is "_______."
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"No $R$ have property $S$."
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3. A negation for "For every $x$, if $x$ has property $P$ then $x$ has property
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$Q$" is "_______."
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"There exists at least one $x$ such that $x$ has property $P$ and $x$ does not
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have property $Q$."
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4. The converse of "For every $x$, if $x$ has property $P$ then $x$ has property
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$Q$" is "_______."
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"For every $x$, if $x$ has property $Q$ then $x$ has property $P$."
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5. The contrapositive of "For every $x$, if $x$ has property $P$ then $x$ has
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property $Q$" is "_______."
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"For every $x$, if $x$ does not have property $Q$, then $x$ does not have
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property $P$."
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6. The inverse of "For every $x$, if $x$ has property $P$ then $x$ has property
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$Q$" is "_______."
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"For every $x$, if $x$ does not have property $P$, then $x$ does not have
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property $Q$."
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---
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**Test Yourself**
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Page 165
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1. To establish the truth of a statement of the form
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"$\forall x \text{ in } D, \exists y \text{ in } E \text{ such that } P(x)$,"
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you imagine that someone has given you an element $x$ from $D$ but that you
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have no control over what that element is. Then you need to find _______ with
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the property that the $x$ the person gave you together with the _______ you
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subsequently found satisfy _______.
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$y \in E$; $y$; $P(x, y)$
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2. To establish the truth of a statement of the form
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"$\exists x \text{ in } D \text{ such that } \forall y \text{ in } E, P(x, y)$,"
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you need to find _______ so that no matter what _______ a person might
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subsequently give you, _______ will be true.
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$x \in D$; $y \in E$; $P(x, y)$
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3. Consider the statement
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"$\forall x, \exists y \text{ such that } P(x, y), \text{ a property involving } x \text{ and } y, \text{ is true}$."
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A negation for this statement is "_______."
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"$\exists x$ such that $\forall y$, the property $P(x, y)$ is false."
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4. Consider the statement
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"$\exists x \text{ such that } \forall y, P(x, y), \text{ a property involving } x \text{ and } y, \text{ is true}$."
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A negation for this statement is "_______."
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"$\forall x, \exists y$ such that the property $P(x, y)$ is false."
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5. Suppose $P(x, y)$ is some property involving $x$ and $y$, and suppose the
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statement
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"$\forall x \text{ in } D, \exists y \text{ in } E \text{ such that } P(x, y)$"
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is true. Then the statement
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"$\exists x \text{ in } D \text{ such that } \forall y \text{ in } E, P(x, y)$"
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a. is true.
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b. is false.
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c. may be true or may be false.
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c is the answer, it may be true or false depending on the nature of the property
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involving $x$, and $y$. In other words, it relies on what $P(x, y)$ states.
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