142 lines
3.9 KiB
Markdown
142 lines
3.9 KiB
Markdown
Page 411$a
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**Test Yourself**
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1. The notation $A \subseteq B$ is read "_____" and means that _____.
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The set $A$ is a subset of the set $B$; if $x \in A$ then $x \in B$
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2. To use an element argument for proving that a set $X$ is a subset of a set
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$Y$, you suppose that _____ and show that _____.
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$x$ is a particular but arbitrarily chosen element of $X$; $x$ is an element of
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$Y$.
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3. To disprove that a set $X$ is a subset of a set $Y$, you show that there is
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_____.
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an element in $X$ that is not in $Y$.
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4. An element $x$ is in $A \cup B$ if, and only if, _____.
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$x$ is in either $A$ or $B$.
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5. An element $x$ is in $A \cap B$ if, and only if, _____.
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$x$ is in both $A$ and $B$.
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6. An element $x$ is in $B - A$ if, and only if, _____.
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$x$ is in $B$ but not in $A$.
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7. An element $x$ is in $A^c$ if, and only if, _____.
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$x$ is in the universal set and is not in $A$.
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8. The empty set is a set with _____.
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no elements.
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9. The power set of a set $A$ is _____.
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the set of all subsets of $A$.
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10. Sets $A$ and $B$ are disjoint if, and only if, _____.
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they have no elements in common, or $A \cap B = \emptyset$.
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11. A collection of nonempty sets $A_1, A_2, A_3, \dots$ is a partition of a set
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$A$ if, and only if, _____.
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all $A_i$ are a subset of $A$, but are also disjoint.
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$A$ is the union of all the sets $A_1, A_2, A_3, \dots$ and
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$A_i \cap A_j = \emptyset$ whenever $i \neq j$.
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---
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Page 426
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**Test Yourself**
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1. To prove that a set $X$ is a subset of a set $A \cap B$, you suppose that $x$
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is any element of $X$ and you show that $x \in A$ _____ $x \in B$.
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and
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2. To prove that a set $X$ is a subset of a set $A \cup B$, you suppose that $x$
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is any element of $X$ and you show that $x \in A$ _____ $x \in B$.
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or
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3. To prove that a set $A \cup B$ is a subset of a set $X$, you start with any
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element $x$ in $A \cup B$ and consider the two cases _____ and _____. You
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then show that in either case _____.
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$x \in A$; $x \in B$; $x \in X$
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4. To prove that a set $A \cap B$ is a subset of $X$, you suppose that _____ and
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you show that _____.
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$x \in A \cap B$; $x \in X$
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5. To prove that a set $X$ equals a set $Y$, you prove that _____ and that
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_____.
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$X \subseteq Y$; $Y \subseteq X$
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6. To prove that a set $X$ does not equal a set $Y$, you need to find an element
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that is in _____ and not _____ or that is in _____ and not _____.
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$X$; in $Y$; $Y$; in $X$
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---
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Page 435
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**Test Yourself**
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1. Given a proposed set identity involving set variables $A$, $B$, and $C$, the
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most common way to show that the equation does not hold in general is to find
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concrete sets $A$, $B$, and $C$ that, when substituted for the set variables
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in the equation, _____.
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the equation does not hold.
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make the left-hand side unequal to the right-hand side
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2. When using the algebraic method for proving a set identity, it is important
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to _____ for every step.
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cite the property from 6.2.2 used
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3. When applying a property from Theorem 6.2.2, it must be used _____ as it is
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stated.
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exactly
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---
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Page 445
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**Test Yourself**
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1. In the comparison between the structure of the set of statement forms and the
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set of subsets of a universal set, the _or_ operation $\vee$ corresponds to
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_____, the _and_ operation $\wedge$ corresponds to _____, a tautology
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$\mathbf{t}$ corresponds to _____, a contradiction $\mathbf{c}$ corresponds
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to _____, and the negation operation, denoted $\neg$, corresponds to _____.
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$\cup$;$\cap$,$U$,$\emptyset$,$^c$
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2. The operations of $+$ and $\cdot$ in a Boolean algebra are generalizations of
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the operations of _____ and _____ in the set of all statement forms in a
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given finite number of variables and the operations of _____ and _____ in the
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set of all subsets of a given set.
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$\vee$;$\wedge$;$\cup$;$\cap$
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3. Russell showed that the following proposed "set definition" could not
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actually define a set: _____.
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the set of all sets that are not elements of themselves
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