Page 458

**Test Yourself**

1. Given a function $f$ from a set $X$ to a set $Y$, $f(x)$ is _____.

the unique output element in $Y$ that is related to $x$ by $f$.

2. Given a function $f$ from a set $X$ to a set $Y$, if $f(x) = y$ then $y$ is
   called _____ or _____ or _____.

the value of $f$ at $x$; the image of $x$ under $f$; the output of $f$ for the
input $x$

3. Given a function $f$ from a set $X$ to a set $Y$, the range of $f$ (or the
   image of $X$ under $f$) is _____.

the set of all $y$ in $Y$ such that $f(x) = y$

4. Given a function $f$ from a set $X$ to $Y$, if $f(x) = y$ then $x$ is called
   _____ or _____.

an inverse image of $y$ under $f$; a preimage of $y$

5. Given a function $f$ from a set $X$ to a set $Y$, if $y \in Y$ then
   $f^{-1}(y) =$ _____ and is called _____.

$\{x \in X | f(x) = y\}$; the inverse image of $y$

6. Given functions $f$ and $g$ from a set $X$ to a set $Y$, $f = g$ if, and only
   if, _____.

$f(x) = g(x)$ for every $x \in X$

7. Given positive real numbers $x$ and $b$ with $b \neq 1$, $\log_b(x) =$ _____.

the exponent to which $b$ must be raised to obtain $x$.

8. Given a function $f$ from a set $X$ to a set $Y$ and a subset $A$ of $X$,
   $f(A) =$ _____.

$\{y \in Y | y = f(x) \text{ for some } x \in A\}$

9. Given a function $f$ from a set $X$ to a set $Y$ and a subset $C$ of $Y$,
   $f^{-1}(C) =$ _____.

$\{x \in X | f(x) \in C\}$

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Page 480

**Test Yourself**

1. If $F$ is a function from a set $X$ to a set $Y$, then $F$ is one-to-one if,
   and only if, _____.

for all $x_1$ and $x_2$ in $X$, if $F(x_1) = F(x_2)$ then $x_1 = x_2$

2. If $F$ is a function from a set $X$ to a set $Y$, then $F$ is not one-to-one
   if, and only if, _____.

for all $x_1$ and $x_2$ in $X$, if $F(x_1) = F(x_2)$ then $x_1 \neq x_2$

3. If $F$ is a function from a set $X$ to a set $Y$, then $F$ is onto if, and
   only if, _____.

for every element $y$ in $Y$, there exists at least one element $x$ in $X$ such
that $f(x) = y$

4. If $F$ is a function from a set $X$ to a set $Y$, then $F$ is not onto if,
   and only if, _____.

for every element $y$ in $Y$, there exists at least one element $x$ in $X$ such
that $f(x) \neq y$

5. The following two statements are _____:

$$ \forall u, v \in U, \text{ if } H(u) = H(v) \text{ then } u = v $$

$$ \forall u, v \in U, \text{ if } u \neq v \text{ then } H(u) \neq H(v) $$

logically equivalent ways of expressing what it means for a function $H$ to be
one-to-one (The second is the contrapositive of the first.)

6. Given a function $F: X \to Y$ where $X$ is an infinite set, to prove that $F$
   is one-to-one, you suppose that _____ and then you show that _____.

$x_1$ and $x_2$ are any _[particular but arbitrarily chosen]_ elements in $X$
with the property that $F(x_1) = F(x_2)$; $x_1 = x_2$

7. Given a function $F: X \to Y$ where $X$ is an infinite set, to prove that $F$
   is onto, you suppose that _____ and then you show that _____.

$y$ is any _[particular but arbitrarily chosen]_ element in $Y$; there exists at
least one element $x$ in $X$ such that $F(x) = y$

8. Given a function $F: X \to Y$, to prove that $F$ is not one-to-one, you
   _____.

show that there are concrete elements $x_1$ and $x_2$ in $X$ with the property
that $F(x_1) = F(x_2)$ and $x_1 \neq x_2$

9. Given a function $F: X \to Y$, to prove that $F$ is not onto, you _____.

show that there is a concrete element $y$ in $Y$ with the property that
$F(x) \neq y$ for any element $x$ in $X$

10. A one-to-one correspondence from a set $X$ to a st $Y$ is a _____ that is
    _____.

function from $X$ to $Y$; both one-to-one and onto

11. If $F$ is a one-to-one correspondence from a set $X$ to a set $Y$ and $y$ is
    in $Y$, then $F^{-1}(y)$ is _____.

the unique element $x$ in $X$ such that $F(x) = y$ (in other words, $F^{-1}(y)$
is the unique preimage of $y$ in $X$)

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Page 494

**Test Yourself**

1. If $f$ is a function from $X$ to $Y'$, $g$ is a function from $Y \to Z$, and
   $Y' \subseteq Y$, then $g \circ f$ is a function from _____ to _____, and
   $(g \circ f)(x) =$ _____ for every $x$ in $X$.

$X$; $Z$, $g(f(x))$

2. If $f$ is a function from $X$ to $Y$ and $I_x$ and $I_y$ are the identity
   functions from $X$ to $X$ and $Y$ to $Y$, respectively, then $f \circ I_x =$
   _____ and $I_y \circ f =$ _____.

$f$; $f$

3. If $f$ is a one-to-one correspondence from $X$ to $Y$, then
   $f^{-1} \circ f =$ _____ and $f \circ f^{-1} =$ _____.

$I_X$; $I_Y$

4. If $f$ is a one-to-one function from $X$ to $Y$ and $g$ is a one-to-one
   function from $Y$ to $Z$, you prove that $g \circ f$ is one-to-one by
   supposing that _____ and then showing that _____.

for some $x_1, x_2 \in X$, $(g \circ f)(x_1) = (g \circ f)(x_2)$; $x_1 = x_2$

5. If $f$ is an onto function from $X$ to $Y$ and $g$ is an onto function from
   $Y$ to $Z$, you prove that $g \circ f$ is onto by supposing that _____ and
   then showing that _____.

for some $z \in Z$, there exists some $x \in X$, such that $(g \circ f)(x) = z$

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Page 507

**Test Yourself**

1. A set is finite if, and only if, _____.

it is the empty set or there is a one-to-one correspondence from
$\{1, 2, \dots n\}$ to it, for some positive integer $n$.

2. To prove that a set $A$ has the same cardinality as a set $B$ you must _____.

show that there is a function one-to-one correspondence from $A$ to $B$.

3. The reflexive property of cardinality says that given any set $A$, _____.

$A$ has the same cardinality as $A$.

4. The symmetric property of cardinality says that given any sets $A$ and $B$,
   _____.

if $A$ has the same cardinality as $B$, then $B$ has the same cardinality as
$A$.

5. The transitive property of cardinality says that given any sets $A$, $B$, and
   $C$, _____.

if $A$ has the same cardinality as $B$, and if $B$ has the same cardinality as
$C$, then $A$ has the same cardinality as $C$.

6. A set is called countably infinite if, and only if, _____.

it has the same cardinality as the set of all positive integers
($\mathbb{Z}^+$).

7. A set is called countable if, and only if, _____.

it is finite or countably infinite

8. In each of the following, fill in the blank with the word _countable_ or the
   word _uncountable_.

a. The set of all integers is _____.

countable

b. The set of all rational numbers is _____.

countable

c. The set of all real numbers between $0$ and $1$ is _____.

uncountable

d. The set of all real numbers is _____.

uncountable

9. The Cantor diagonalization process is used to prove that _____.

the set of all real numbers between $0$ and $1$ is uncountable