Difference between revisions of "Sets:Operations"
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==Definitions== | ==Definitions== | ||
The two main set operations that we deal with are union and intersection. The union of two sets <math> A </math> and <math> B </math> is defined as <math> A \cup B = \{x : x\in A </math> or <math> x\in B\} </math>. For example: | The two main set operations that we deal with are union and intersection. The union of two sets <math> A </math> and <math> B </math> is defined as <math> A \cup B = \{x : x\in A </math> or <math> x\in B\} </math>. For example: | ||
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* The union of <math> A = \{1, 3, 5, 7, 9\} </math> and <math> B = \{0, 1, 2, 4, 6, 8, 9\} </math> is <math> A \cup B = \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\} </math> | * The union of <math> A = \{1, 3, 5, 7, 9\} </math> and <math> B = \{0, 1, 2, 4, 6, 8, 9\} </math> is <math> A \cup B = \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\} </math> | ||
* The union of the even integers and odd integers is <math> \Z </math>. | * The union of the even integers and odd integers is <math> \Z </math>. | ||
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* For sets <math> A </math> and <math> B </math> such that <math> A \subseteq B </math>, <math> A \cup B = B </math>, since all elements of <math> A </math> are already in <math> B </math> if <math> A \subseteq B </math>. | * For sets <math> A </math> and <math> B </math> such that <math> A \subseteq B </math>, <math> A \cup B = B </math>, since all elements of <math> A </math> are already in <math> B </math> if <math> A \subseteq B </math>. | ||
| − | The intersection of <math> A </math> and <math> B </math> is defined as <math> A \cap B = \{x : x\in A </math> and <math> x\in B\} </math>; that is, the intersection of <math> A </math> and <math> B </math> is the set of all elements shared by the two sets. | + | The intersection of <math> A </math> and <math> B </math> is defined as <math> A \cap B = \{x : x\in A </math> and <math> x\in B\} </math>; that is, the intersection of <math> A </math> and <math> B </math> is the set of all elements shared by the two sets. Sets <math> A </math> and <math> B </math> are "disjoint" if <math> A \cap B = \empty </math>. |
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* The intersection of <math> A = \{1, 3, 5, 7, 9\} </math> and <math> B = \{0, 1, 2, 4, 6, 8, 9\} </math> is <math> A \cap B = \{1, 9\} </math>. | * The intersection of <math> A = \{1, 3, 5, 7, 9\} </math> and <math> B = \{0, 1, 2, 4, 6, 8, 9\} </math> is <math> A \cap B = \{1, 9\} </math>. | ||
* The intersection of the even integers and odd integers is the empty set, since no element between these two sets is shared (an integer cannot be both even and odd). | * The intersection of the even integers and odd integers is the empty set, since no element between these two sets is shared (an integer cannot be both even and odd). | ||
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* For sets <math> A </math> and <math> B </math> such that <math> A \subseteq B </math>, <math> A \cup B = A </math>. | * For sets <math> A </math> and <math> B </math> such that <math> A \subseteq B </math>, <math> A \cup B = A </math>. | ||
| − | + | There are a few other common set operations. The set difference of <math> A </math> and <math> B </math> is defined as <math> A\backslash B = \{x : x\in A, x\not\in B\} </math>. We read <math> A\backslash B </math> (also sometimes denoted as <math> A-B </math>) as "<math> A </math> without <math> B </math>". Note that this operation is not commutative; that is, <math> A\backslash B </math> does not equal <math> B\backslash A </math> in most cases. Example: if <math> A = \{1, 2, 3, 4, 5\} </math> and <math> B = \{0, 1, 3, 4, 5, 6\} </math>, then <math> A\backslash B = \{2\} </math> and <math> B\backslash A = \{0, 6\} </math>. | |
==Resources== | ==Resources== | ||
* [https://link-springer-com.libweb.lib.utsa.edu/content/pdf/10.1007%2F978-1-4419-7127-2.pdf Course Textbook], pages 101-115 | * [https://link-springer-com.libweb.lib.utsa.edu/content/pdf/10.1007%2F978-1-4419-7127-2.pdf Course Textbook], pages 101-115 | ||
Revision as of 15:53, 26 September 2021
Definitions
The two main set operations that we deal with are union and intersection. The union of two sets Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A } and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B } is defined as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A \cup B = \{x : x\in A } or . For example:
- The union of and is
- The union of the even integers and odd integers is .
- The union of the set of rational numbers and the set of irrational numbers is .
- , and .
- For sets Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A } and such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A \subseteq B } , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A \cup B = B } , since all elements of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A } are already in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B } if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A \subseteq B } .
The intersection of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A } and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B } is defined as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A \cap B = \{x : x\in A } and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x\in B\} } ; that is, the intersection of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A } and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B } is the set of all elements shared by the two sets. Sets Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A } and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B } are "disjoint" if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A \cap B = \empty } .
- The intersection of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A = \{1, 3, 5, 7, 9\} } and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B = \{0, 1, 2, 4, 6, 8, 9\} } is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A \cap B = \{1, 9\} } .
- The intersection of the even integers and odd integers is the empty set, since no element between these two sets is shared (an integer cannot be both even and odd).
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A \cap A = A } , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \empty\cap A = \empty } .
- For sets Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A } and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B } such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A \subseteq B } , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A \cup B = A } .
There are a few other common set operations. The set difference of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A } and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B } is defined as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A\backslash B = \{x : x\in A, x\not\in B\} } . We read Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A\backslash B } (also sometimes denoted as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A-B } ) as "Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A } without Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B } ". Note that this operation is not commutative; that is, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A\backslash B } does not equal Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B\backslash A } in most cases. Example: if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A = \{1, 2, 3, 4, 5\} } and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B = \{0, 1, 3, 4, 5, 6\} } , then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A\backslash B = \{2\} } and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B\backslash A = \{0, 6\} } .
Resources
- Course Textbook, pages 101-115
