Difference between revisions of "Subset"
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| − | We say a [[set]] <math>A</math> is a '''subset''' of another set <math>B</math> if every [[element]] of <math>A</math> is also an element of <math>B</math>, and we denote this by <math>A \sub B</math>. The [[empty set]] is a subset of every set, and every set is a subset of itself. The notation <math>A \subseteq B</math> emphasizes that <math>A</math> may be equal to <math>B</math>, while <math> | + | We say a [[set]] <math>A</math> is a '''subset''' of another set <math>B</math> if every [[element]] of <math>A</math> is also an element of <math>B</math>, and we denote this by <math>A \sub B</math>. The [[empty set]] is a subset of every set, and every set is a subset of itself. The notation <math>A \subseteq B</math> emphasizes that <math>A</math> may be equal to <math>B</math>, while <math>A \subsetneq B</math> says that <math>A</math> is any subset of <math>B</math> other than <math>B</math> itself. |
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=== Intermediate === | === Intermediate === | ||
* [[1992_AIME_Problems/Problem_2 | 1992 AIME Problem 2]] | * [[1992_AIME_Problems/Problem_2 | 1992 AIME Problem 2]] | ||
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| + | [[Category:Set Theory]] | ||
Revision as of 11:10, 3 December 2007
We say a set 
is a subset of another set 
if every element of is also an element of , and we denote this by $A \sub B$ (Error compiling LaTeX. Unknown error_msg). The empty set is a subset of every set, and every set is a subset of itself. The notation 
emphasizes that may be equal to , while 
says that is any subset of other than itself.
The following is a true statement:
$\emptyset \sub \{1, 2\} \sub \mathbb{N} \sub \mathbb{Z} \sub \mathbb{Q} \sub \mathbb{R} \sub \mathbb{C} \sub \mathbb{C}\, \cup\{\textrm{Groucho, Harpo, Chico}\} \supset \{1, 2, i, \textrm{Groucho}\}$ (Error compiling LaTeX. Unknown error_msg)
The set of all subsets of a given set 
is called the power set of and is denoted 
or 
.
