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 $X$)