| Version 6 |
Version 5 |
| The \emph{union} of two sets $A$ and $B$ is the set which contains all $x \in A$ and all $x \in B$, denoted $A \cup B$. We can extend this to any (finite or infinite) family $(A_i)_{i\in I}$, writing $\bigcup_{i\in I}A_i$ for the union of this family. Formally, for a family $(A_i)_{i\in I}$ of sets: |
The \emph{union} of two sets $A$ and $B$ is the set which contains all $x \in A$ and all $x \in B$, denoted $A \cup B$. We can extend this to any (finite or infinite) family $(A_i)_{i\in I}$, writing $\bigcup_{i\in I}A_i$ for the union of this family. Formally, for a family $(A_i)_{i\in I}$ of sets: |
| \[ x \in \bigcup_{i\in I}A_i\; \Leftrightarrow \;\bigvee_{i\in I}\, (x\in A_i) \] |
\[ x \in \bigcup_{i\in I}A_i\; \Leftrightarrow \;\bigvee_{i\in I}\, (x\in A_i) \] |
| Alternatively, and equivalently, |
Alternatively, and equivalently, |
| \[x \in \bigcup_{i\in I}A_i\; \Leftrightarrow \;\exists i\in I\text{ such that } x\in A_i\] |
\[x \in \bigcup_{i\in I}A_i\; \Leftrightarrow \;\exists i\in I\text{ such that } x\in A_i\] |
| This characterization makes it much clearer that if $I$ is itself the empty set (that is, if we are taking the union of an empty family), then the union is empty; that is, |
This characterization makes it much clearer that if $I$ is itself the empty set (that is, if we are taking the union of an empty family), then the union is empty; that is, |
| \[\bigcup_{i\in\emptyset}A_i=\emptyset\] |
\[\bigcup_{i\in\emptyset}A_i=\emptyset\] |
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| Often elements of sets are taken from some universe $U$ of elements under consideration (for example, the real numbers $\Reals$, or living things on the planet, or words in a particular book). When this is the case, it is meaningful to discuss the \emph{complement} of a set: if $A$ is a set of elements from some universe $U$, then the complement of $A$ is the set |
Often elements of sets are taken from some universe $U$ of elements under consideration (for example, the real numbers $\Reals$, or living things on the planet, or words in a particular book). When this is the case, it is meaningful to discuss the \emph{complement} of a set: if $A$ is a set of elements from some universe $U$, then the complement of $A$ is the set |
| \[A^C = U\backslash A= \{x\in U\suchthat x\notin A\}\] |
\[A^C = U\backslash A= \{x\in U\suchthat x\notin A\}\] |
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| From an axiomatic point of view, the existence of the union is guaranteed by the axiom of union. |
From an axiomatic point of view, the existence of the union is guaranteed by the axiom of union. |
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| Note that the sets $A_i$ may be, but need not be, disjoint. Unions satisfy some basic properties that are obvious from the definitions: |
Note that the sets $A_i$ may be, but need not be, disjoint. Unions satisfy some basic properties that are obvious from the definitions: |
| \begin{itemize} |
\begin{itemize} |
| \item Idempotency: $A \cup A = A$ |
\item Idempotency: $A \cup A = A$ |
| \item $A \cup A^C = U$ where $U$ is the \emph{universe} of $A$ |
\item $A \cup A^C = U$ where $U$ is the \emph{universe} of $A$ |
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\item Commutativity: $A \cup B = B \cup A$
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\item Symmetry: $A \cup B = B \cup A$
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| \item Associativity: $(A \cup B) \cup C = A \cup (B \cup C)$ |
\item Associativity: $(A \cup B) \cup C = A \cup (B \cup C)$ |
| \end{itemize} |
\end{itemize} |
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| Here are some examples of set unions: |
Here are some examples of set unions: |
| \begin{gather*} |
\begin{gather*} |
| \{1,2\}\cup\{3,4\} = \{1,2,3,4\}\\ |
\{1,2\}\cup\{3,4\} = \{1,2,3,4\}\\ |
| \{1,2\}\cup\{1,4\} = \{1,2,4\}\\ |
\{1,2\}\cup\{1,4\} = \{1,2,4\}\\ |
| \{blue, green\}\cup\emptyset = \{blue, green\}\\ |
\{blue, green\}\cup\emptyset = \{blue, green\}\\ |
| \{x\in\Ints\ \mid\ x\geq 1\}\cup\{x\in\Ints\ \mid\ x\leq\-1\} = \{x\in\Ints\ \mid\ x\neq 0\}\\ |
\{x\in\Ints\ \mid\ x\geq 1\}\cup\{x\in\Ints\ \mid\ x\leq\-1\} = \{x\in\Ints\ \mid\ x\neq 0\}\\ |
| \{x\in\Reals\ \mid\ x\geq 1\}\cup\{x\in\Reals\ \mid\ x\leq\-1\} = \{x\in\Reals\ \mid\ -1<x<1\} = (-1,1)\\ |
\{x\in\Reals\ \mid\ x\geq 1\}\cup\{x\in\Reals\ \mid\ x\leq\-1\} = \{x\in\Reals\ \mid\ -1<x<1\} = (-1,1)\\ |
| \{x\in\Reals\ \mid\ x\geq 2\}\cup\{x\in\Reals\ \mid\ x\leq 2\} = \Reals\\ |
\{x\in\Reals\ \mid\ x\geq 2\}\cup\{x\in\Reals\ \mid\ x\leq 2\} = \Reals\\ |
| \bigcup_{\substack{n\in\Ints\\n>0}} \{x=p/q\in\Rats\ \mid\ q<n\text{ when }p/q\text{ is in lowest terms }\} = \Rats |
\bigcup_{\substack{n\in\Ints\\n>0}} \{x=p/q\in\Rats\ \mid\ q<n\text{ when }p/q\text{ is in lowest terms }\} = \Rats |
| \end{gather*} |
\end{gather*} |
| The first, third, fourth and fifth of these are the union of disjoint sets, while the second, sixth and seventh are not - in those cases, the sets overlap each other. |
The first, third, fourth and fifth of these are the union of disjoint sets, while the second, sixth and seventh are not - in those cases, the sets overlap each other. |
