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 \ | + | 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 \subset 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. In the latter case, <math>A</math> is called a ''proper subset''. |
The following is a true statement: | The following is a true statement: | ||
− | <math>\emptyset \ | + | <math>\emptyset \subset \{1, 2\} \subset \mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C} \subset \mathbb{C}\cup\{\textrm{Groucho, Harpo, Chico}\} \supset \{1, 2, i, \textrm{Groucho}\}</math> |
− | The set of all subsets of a given set <math>S</math> is called the [[power set]] of <math>S</math> and is denoted <math>\mathcal{P}(S)</math> or <math>2^S</math>. | + | The set of all subsets of a given set <math>S</math> is called the [[power set]] of <math>S</math> and is denoted <math>\mathcal{P}(S)</math> or <math>2^S</math>. The number of subsets of <math>S</math> is <math>2^{|S|}</math>. |
== Example Problems == | == Example Problems == | ||
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=== Intermediate === | === Intermediate === | ||
* [[1992_AIME_Problems/Problem_2 | 1992 AIME Problem 2]] | * [[1992_AIME_Problems/Problem_2 | 1992 AIME Problem 2]] | ||
+ | |||
+ | [[Category:Set theory]] |
Latest revision as of 09:32, 13 August 2011
We say a set is a subset of another set
if every element of
is also an element of
, and we denote this by
. 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. In the latter case,
is called a proper subset.
The following is a true statement:
The set of all subsets of a given set is called the power set of
and is denoted
or
. The number of subsets of
is
.