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1. If $F$ is a function from a set $X$ to a set $Y$, then $F$ is one-to-one if, 1. If $F$ is a function from a set $X$ to a set $Y$, then $F$ is one-to-one if,
and only 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 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, _____. 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 3. If $F$ is a function from a set $X$ to a set $Y$, then $F$ is onto if, and
only if, _____. 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, 4. If $F$ is a function from a set $X$ to a set $Y$, then $F$ is not onto if,
and only 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 _____: 5. The following two statements are _____:
$$ \forall u, v \in U, \text{ if } H(u) = H(v) \text{ then } u = v $$ $$ \forall u, v \in U, \text{ if } H(u) = H(v) \text{ then } u = v $$
$$ \forall u, v \in U, \text{ if } u \neq v \text{ then } H(u) \neq H(v) $$ $$ \forall u, v \in U, \text{ if } u \neq v \text{ then } H(u) \neq H(v) $$
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 $F$ 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 _____. 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 $F$ 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 _____. 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 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 _____. 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 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 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 _____. 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 $X$)