Summary

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Section Summary

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Important ideas that we discussed in this section include the following.

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A function $f$ from a nonempty set $A$ to a set $B$ is a collection of ordered pairs $(a, b)$ so that for each $a \in A$ there is a pair $(a, b)$ in $f,$ and if $(a, b)$ and $(a, b^{'})$ are in $f,$ then $b = b^{'} .$ If $f$ is a function we use the notation $f (a) = b$ to indicate that $(a, b) \in f .$
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If $f$ is a function from $A$ to $B,$ the set $A$ is the domain of the function.
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If $f$ is a function from $A$ to $B,$ the set $B$ is the codomain of the function. The set

${f (a) ∣ a \in A}$

is the range of the function. So the range of a function is a subset of the codomain.
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A function $f$ from a set $A$ to a set $B$ is an injection if, whenever $f (a) = f (a^{'})$ for $a,$ $a^{'} \in A,$ then $a = a^{'} .$ The function $f$ is a surjection if, whenever $b \in B,$ then there is an $a \in A$ so that $f (a) = b .$
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If $f$ is a function from a set $A$ to a set $B$ and if $g$ is a function from $B$ to a set $C,$ then the composite $g \circ f$ is a function from $A$ to $C$ defined by $(g \circ f) (a) = g (f (a))$ for every $a \in A .$
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A function $f$ from a set $A$ to a set $B$ is a bijection if $f$ is both a surjection and injection. When $f$ is a bijection from $A$ to $B,$ then $f$ has an inverse $f^{- 1}$ defined by $f^{- 1} (b) = a$ when $f (a) = b .$
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If $f$ is a function from a set $A$ to a set $B,$ and if $C$ is a subset of $A,$ then image of $C$ under $f$ is the set

$f (C) = {f (c) ∣ c \in C},$

and if $D$ is a subset of $Y,$ the inverse image of $D$ is the set

$f^{- 1} (D) = {a \in A ∣ f (a) \in D} .$
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Important properties that relate images and inverse images of sets and set unions are the following. If $f$ is a function from a set $X$ to a set $Y,$ and if ${A_{α}}$ is a collection of subsets of $X$ for $α$ in some indexing set $I,$ and ${B_{β}}$ be a collection of subsets of $Y$ for $β$ in some indexing set $J,$ then
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  $f (⋃_{α \in I} A_{α}) = ⋃_{α \in I} f (A_{α})$ and
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  $f^{- 1} (⋃_{β \in J} B_{β}) = ⋃_{β \in J} f^{- 1} (B_{β}) .$

Topology: An Inquiry-Based Approach

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Section Summary