153 lines
4.9 KiB
Markdown
153 lines
4.9 KiB
Markdown
Page 458
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**Test Yourself**
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1. Given a function $f$ from a set $X$ to a set $Y$, $f(x)$ is _____.
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the unique output element in $Y$ that is related to $x$ by $f$.
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2. Given a function $f$ from a set $X$ to a set $Y$, if $f(x) = y$ then $y$ is
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called _____ or _____ or _____.
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the value of $f$ at $x$; the image of $x$ under $f$; the output of $f$ for the
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input $x$
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3. Given a function $f$ from a set $X$ to a set $Y$, the range of $f$ (or the
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image of $X$ under $f$) is _____.
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the set of all $y$ in $Y$ such that $f(x) = y$
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4. Given a function $f$ from a set $X$ to $Y$, if $f(x) = y$ then $x$ is called
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_____ or _____.
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an inverse image of $y$ under $f$; a preimage of $y$
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5. Given a function $f$ from a set $X$ to a set $Y$, if $y \in Y$ then
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$f^{-1}(y) =$ _____ and is called _____.
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$\{x \in X | f(x) = y\}$; the inverse image of $y$
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6. Given functions $f$ and $g$ from a set $X$ to a set $Y$, $f = g$ if, and only
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if, _____.
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$f(x) = g(x)$ for every $x \in X$
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7. Given positive real numbers $x$ and $b$ with $b \neq 1$, $\log_b(x) =$ _____.
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the exponent to which $b$ must be raised to obtain $x$.
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8. Given a function $f$ from a set $X$ to a set $Y$ and a subset $A$ of $X$,
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$f(A) =$ _____.
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$\{y \in Y | y = f(x) \text{ for some } x \in A\}$
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9. Given a function $f$ from a set $X$ to a set $Y$ and a subset $C$ of $Y$,
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$f^{-1}(C) =$ _____.
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$\{x \in X | f(x) \in C\}$
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---
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Page 480
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**Test Yourself**
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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,
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and only if, _____.
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for all $x_1$ and $x_2$ in $X$, if $F(x_1) = F(x_2)$ then $x_1 = x_2$
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2. If $F$ is a function from a set $X$ to a set $Y$, then $F$ is not one-to-one
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if, and only if, _____.
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for all $x_1$ and $x_2$ in $X$, if $F(x_1) = F(x_2)$ then $x_1 \neq x_2$
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3. If $F$ is a function from a set $X$ to a set $Y$, then $F$ is onto if, and
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only if, _____.
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for every element $y$ in $Y$, there exists at least one element $x$ in $X$ such
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that $f(x) = y$
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4. If $F$ is a function from a set $X$ to a set $Y$, then $F$ is not onto if,
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and only if, _____.
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for every element $y$ in $Y$, there exists at least one element $x$ in $X$ such
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that $f(x) \neq y$
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5. The following two statements are _____:
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$$ \forall u, v \in U, \text{ if } H(u) = H(v) \text{ then } u = v $$
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$$ \forall u, v \in U, \text{ if } u \neq v \text{ then } H(u) \neq H(v) $$
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logically equivalent ways of expressing what it means for a function $H$ to be
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one-to-one (The second is the contrapositive of the first.)
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6. Given a function $F: X \to Y$ where $X$ is an infinite set, to prove that $F$
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is one-to-one, you suppose that _____ and then you show that _____.
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$x_1$ and $x_2$ are any _[particular but arbitrarily chosen]_ elements in $X$
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with the property that $F(x_1) = F(x_2)$; $x_1 = x_2$
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7. Given a function $F: X \to Y$ where $X$ is an infinite set, to prove that $F$
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is onto, you suppose that _____ and then you show that _____.
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$y$ is any _[particular but arbitrarily chosen]_ element in $Y$; there exists at
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least one element $x$ in $X$ such that $F(x) = y$
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8. Given a function $F: X \to Y$, to prove that $F$ is not one-to-one, you
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_____.
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show that there are concrete elements $x_1$ and $x_2$ in $X$ with the property
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that $F(x_1) = F(x_2)$ and $x_1 \neq x_2$
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9. Given a function $F: X \to Y$, to prove that $F$ is not onto, you _____.
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show that there is a concrete element $y$ in $Y$ with the property that
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$F(x) \neq y$ for any element $x$ in $X$
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10. A one-to-one correspondence from a set $X$ to a st $Y$ is a _____ that is
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_____.
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function from $X$ to $Y$; both one-to-one and onto
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11. If $F$ is a one-to-one correspondence from a set $X$ to a set $Y$ and $y$ is
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in $Y$, then $F^{-1}(y)$ is _____.
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the unique element $x$ in $X$ such that $F(x) = y$ (in other words, $F^{-1}(y)$
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is the unique preimage of $y$ in $X$)
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---
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Page 494
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**Test Yourself**
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1. If $f$ is a function from $X$ to $Y'$, $g$ is a function from $Y \to Z$, and
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$Y' \subseteq Y$, then $g \circ f$ is a function from _____ to _____, and
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$(g \circ f)(x) =$ _____ for every $x$ in $X$.
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$X$; $Z$, $g(f(x))$
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2. If $f$ is a function from $X$ to $Y$ and $I_x$ and $I_y$ are the identity
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functions from $X$ to $X$ and $Y$ to $Y$, respectively, then $f \circ I_x =$
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_____ and $I_y \circ f =$ _____.
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$f$; $f$
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3. If $f$ is a one-to-one correspondence from $X$ to $Y$, then
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$f^{-1} \circ f =$ _____ and $f \circ f^{-1} =$ _____.
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$I_X$; $I_Y$
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4. If $f$ is a one-to-one function from $X$ to $Y$ and $g$ is a one-to-one
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function from $Y$ to $Z$, you prove that $g \circ f$ is one-to-one by
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supposing that _____ and then showing that _____.
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for some $x_1, x_2 \in X$, $(g \circ f)(x_1) = (g \circ f)(x_2)$; $x_1 = x_2$
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5. If $f$ is an onto function from $X$ to $Y$ and $g$ is an onto function from
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$Y$ to $Z$, you prove that $g \circ f$ is onto by supposing that _____ and
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then showing that _____.
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for some $z \in Z$, there exists some $x \in X$, such that $(g \circ f)(x) = z$
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