Sets:Operations - Revision history

Khanh at 02:27, 5 February 2022

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Lila at 22:16, 26 September 2021

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Lila at 22:02, 26 September 2021

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Lila: /* Definitions */

2021-09-26T21:53:36Z

Definitions

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Lila: /* Definitions */

2021-09-26T18:46:58Z

Definitions

Lila: Created page with "==Definitions== The two main set operations that we deal with are union and intersection. The union of two sets $A$ and $B$ is defined as $A \c..."$

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Created page with "==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 \c..."

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==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 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 set of rational numbers and the set of irrational numbers is <math> \R </math>.
* <math> A \cup A = A </math>, and <math> \empty\cup A = A </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. For example:
* 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).
* <math> A \cap A = A </math>, and <math> \empty\cap A = \empty </math>.
* For sets <math> A </math> and <math> B </math> such that <math> A \subseteq B </math>, <math> A \cup B = A </math>.

==Resources==
* [ Course Textbook], pages 101-115

@@ Line 17: / Line 17: @@
 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>.
-Another set operation is the Cartesian product (or the product of two sets). The Cartesian product of two sets <math> A </math> and <math> B </math> is defined as <math> A\times B = \{(a, b): a\in A </math> and <math> b\in B\} </math>, where <math> (a,b) </math> is an ordered pair. For example, if <math> A = \{1, 2, 3\} </math> and <math> B = \{0, 1\} </math>, then <math> A\times B = \{(1,0), (1,1), (2,0), (2,1), (3,0), (3, 1)\} </math>. The number of elements in the Cartesian product <math> A\times B </math> is the product of the number of elements in set <math> A </math> and set <math> B </math> (for example, if <math> A </math> has 3 elements and <math> B </math> has 2, then <math> A\times B </math> has 6).
+Another set operation is the Cartesian product (or the product of two sets). The Cartesian product of two sets <math> A </math> and <math> B </math> is defined as <math> A\times B = \{(a, b): a\in A </math> and <math> b\in B\} </math>, where <math> (a,b) </math> is an ordered pair. For example, if <math> A = \{1, 2, 3\} </math> and <math> B = \{0, 1\} </math>, then <math> A\times B = \{(1,0), (1,1), (2,0), (2,1), (3,0), (3, 1)\} </math>. If <math> A </math> and <math> B </math> are both finite sets, then the number of elements in <math> A\times B </math> is the product of the number of elements in set <math> A </math> and set <math> B </math> (for example, if <math> A </math> has 3 elements and <math> B </math> has 2, then <math> A\times B </math> has 6).
 ==Resources==
 * [https://link-springer-com.libweb.lib.utsa.edu/content/pdf/10.1007%2F978-1-4419-7127-2.pdf Course Textbook], pages 101-115

@@ Line 16: / Line 16: @@
 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==
 * [https://link-springer-com.libweb.lib.utsa.edu/content/pdf/10.1007%2F978-1-4419-7127-2.pdf Course Textbook], pages 101-115

← Older revision		Revision as of 21:22, 26 September 2021
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	==Resources==		==Resources==
−	* [ 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

@@ Line 1: / Line 1: @@
 ==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 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>.
@@ Line 7: / Line 8: @@
 * 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. For example:
+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>.
 * 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).
@@ Line 13: / Line 15: @@
 * For sets <math> A </math> and <math> B </math> such that <math> A \subseteq B </math>, <math> A \cup B = A </math>.
-Sets <math> A </math> and <math> B </math> are "disjoint" if <math> A \cap B = \empty </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==
 * [https://link-springer-com.libweb.lib.utsa.edu/content/pdf/10.1007%2F978-1-4419-7127-2.pdf Course Textbook], pages 101-115

Sets:Operations - Revision history

Khanh at 02:27, 5 February 2022

Lila at 22:16, 26 September 2021

Lila at 22:02, 26 September 2021

Lila: /* Definitions */

Lila: /* Resources */

Lila: /* Definitions */

Lila: Created page with "==Definitions== The two main set operations that we deal with are union and intersection. The union of two sets A and B is defined as A \c..."

Lila: Created page with "==Definitions== The two main set operations that we deal with are union and intersection. The union of two sets $A$ and $B$ is defined as $A \c..."$