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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>\displaystyle 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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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:
  
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<math>\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}\}</math>
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<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>
  
  
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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>.
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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]]
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[[Category:Set theory]]

Latest revision as of 08:32, 13 August 2011

We say a set $A$ is a subset of another set $B$ if every element of $A$ is also an element of $B$, and we denote this by $A \subset B$. The empty set is a subset of every set, and every set is a subset of itself. The notation $A \subseteq B$ emphasizes that $A$ may be equal to $B$, while $A \subsetneq B$ says that $A$ is any subset of $B$ other than $B$ itself. In the latter case, $A$ is called a proper subset.


The following is a true statement:

$\emptyset \subset \{1, 2\} \subset \mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C} \subset \mathbb{C}\cup\{\textrm{Groucho, Harpo, Chico}\} \supset \{1, 2, i, \textrm{Groucho}\}$


The set of all subsets of a given set $S$ is called the power set of $S$ and is denoted $\mathcal{P}(S)$ or $2^S$. The number of subsets of $S$ is $2^{|S|}$.

Example Problems

Introductory

Intermediate

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