Page 411$a

**Test Yourself**

1. The notation $A \subseteq B$ is read "_____" and means that _____.

The set $A$ is a subset of the set $B$; if $x \in A$ then $x \in B$

2. To use an element argument for proving that a set $X$ is a subset of a set
   $Y$, you suppose that _____ and show that _____.

$x$ is a particular but arbitrarily chosen element of $X$; $x$ is an element of
$Y$.

3. To disprove that a set $X$ is a subset of a set $Y$, you show that there is
   _____.

an element in $X$ that is not in $Y$.

4. An element $x$ is in $A \cup B$ if, and only if, _____.

$x$ is in either $A$ or $B$.

5. An element $x$ is in $A \cap B$ if, and only if, _____.

$x$ is in both $A$ and $B$.

6. An element $x$ is in $B - A$ if, and only if, _____.

$x$ is in $B$ but not in $A$.

7. An element $x$ is in $A^c$ if, and only if, _____.

$x$ is in the universal set and is not in $A$.

8. The empty set is a set with _____.

no elements.

9. The power set of a set $A$ is _____.

the set of all subsets of $A$.

10. Sets $A$ and $B$ are disjoint if, and only if, _____.

they have no elements in common, or $A \cap B = \emptyset$.

11. A collection of nonempty sets $A_1, A_2, A_3, \dots$ is a partition of a set
    $A$ if, and only if, _____.

all $A_i$ are a subset of $A$, but are also disjoint.

$A$ is the union of all the sets $A_1, A_2, A_3, \dots$ and
$A_i \cap A_j = \emptyset$ whenever $i \neq j$.

---

Page 426

**Test Yourself**

1. To prove that a set $X$ is a subset of a set $A \cap B$, you suppose that $x$
   is any element of $X$ and you show that $x \in A$ _____ $x \in B$.

and

2. To prove that a set $X$ is a subset of a set $A \cup B$, you suppose that $x$
   is any element of $X$ and you show that $x \in A$ _____ $x \in B$.

or

3. To prove that a set $A \cup B$ is a subset of a set $X$, you start with any
   element $x$ in $A \cup B$ and consider the two cases _____ and _____. You
   then show that in either case _____.

$x \in A$; $x \in B$; $x \in X$

4. To prove that a set $A \cap B$ is a subset of $X$, you suppose that _____ and
   you show that _____.

$x \in A \cap B$; $x \in X$

5. To prove that a set $X$ equals a set $Y$, you prove that _____ and that
   _____.

$X \subseteq Y$; $Y \subseteq X$

6. To prove that a set $X$ does not equal a set $Y$, you need to find an element
   that is in _____ and not _____ or that is in _____ and not _____.

$X$; in $Y$; $Y$; in $X$

---

Page 435

**Test Yourself**

1. Given a proposed set identity involving set variables $A$, $B$, and $C$, the
   most common way to show that the equation does not hold in general is to find
   concrete sets $A$, $B$, and $C$ that, when substituted for the set variables
   in the equation, _____.

the equation does not hold.

make the left-hand side unequal to the right-hand side

2. When using the algebraic method for proving a set identity, it is important
   to _____ for every step.

cite the property from 6.2.2 used

3. When applying a property from Theorem 6.2.2, it must be used _____ as it is
   stated.

exactly

---

Page 445

**Test Yourself**

1. In the comparison between the structure of the set of statement forms and the
   set of subsets of a universal set, the _or_ operation $\vee$ corresponds to
   _____, the _and_ operation $\wedge$ corresponds to _____, a tautology
   $\mathbf{t}$ corresponds to _____, a contradiction $\mathbf{c}$ corresponds
   to _____, and the negation operation, denoted $\neg$, corresponds to _____.

$\cup$;$\cap$,$U$,$\emptyset$,$^c$

2. The operations of $+$ and $\cdot$ in a Boolean algebra are generalizations of
   the operations of _____ and _____ in the set of all statement forms in a
   given finite number of variables and the operations of _____ and _____ in the
   set of all subsets of a given set.

$\vee$;$\wedge$;$\cup$;$\cap$

3. Russell showed that the following proposed "set definition" could not
   actually define a set: _____.

the set of all sets that are not elements of themselves