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**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 $i \neq j$.