Summary

Section Summary

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

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We can consider a set to be a well-defined collection of elements.
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A subset of a set is any collection of elements from that set. That is, a subset $S$ of a set $X$ is a set with the property that if $s \in S,$ then $s \in X .$
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If $X$ and $Y$ are sets, then the union $X \cup Y$ is the set

$X \cup Y = {z ∣ z \in X or z \in Y} .$

The union of an arbitrary collection ${X_{α}}$ of sets for $α$ in some indexing set $I$ is the set

$⋃_{α \in I} X_{α} = {z ∣ z \in X_{β} for some β \in I} .$
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If $X$ and $Y$ are sets, then the intersection $X \cap Y$ is the set

$X \cap Y = {z ∣ z \in X and z \in Y} .$

The intersection of an arbitrary collection ${X_{α}}$ of sets for $α$ in some indexing set $I$ is the set

$⋂_{α \in I} X_{α} = {z ∣ z \in X_{β} for all β \in I} .$
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If $X$ is a set and $A$ is a subset of $X,$ then the complement of $A$ in $X$ is the set

$A^{c} = {x \in X ∣ x \notin A} .$
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If ${X_{i}}$ is a collection of sets with $i$ in some indexing set $I,$ where $I$ is finite or $I$ is the set of positive integers, the Cartesian product $Π_{i \in I} X_{i}$ of the sets $X_{i}$ as the set of all ordered tuples of the form $(x_{i})$ where $i \in I .$