Difference between revisions of "Intersection"

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*For any sets <math>A, B</math>, <math>A \cap B \subseteq A</math> and <math>A \cap B \subseteq B</math>.  Thus <math>A \cap B = A</math> if and only if <math>A \subseteq B</math>.
 
*For any sets <math>A, B</math>, <math>A \cap B \subseteq A</math> and <math>A \cap B \subseteq B</math>.  Thus <math>A \cap B = A</math> if and only if <math>A \subseteq B</math>.
  
 
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==Geometrical definition==
 
In [[geometry]], a [[line]] may be considered to be a set of [[point]]s with a particular property (the property of being on that line).  Then the intersection of two lines reduces to the set definition of intersection.  This also extends to other curves and surfaces.
 
In [[geometry]], a [[line]] may be considered to be a set of [[point]]s with a particular property (the property of being on that line).  Then the intersection of two lines reduces to the set definition of intersection.  This also extends to other curves and surfaces.
  

Revision as of 00:10, 15 November 2007

Intersection is a property of multiple sets.


Definition

The intersection of two or more sets is the set of elements which are common to all of them. Thus, the intersection of the sets $\{1, 2, 3\}$ and $\{1, 3, 5\}$ is the set $\{1, 3\}$. The intersection of two or more sets is denoted by the symbol $\cap$, so the preceding example could be written $\{1, 2, 3\} \cap \{1, 3, 5\} = \{1, 3\}$.

Notation

The intersection of $A$ and $B$ is denoted by $A \cap B$

Properties

  • For any sets $A, B$, $A \cap B \subseteq A$ and $A \cap B \subseteq B$. Thus $A \cap B = A$ if and only if $A \subseteq B$.

Geometrical definition

In geometry, a line may be considered to be a set of points with a particular property (the property of being on that line). Then the intersection of two lines reduces to the set definition of intersection. This also extends to other curves and surfaces.

Especially in the geometric context, two objects are said to intersect if their intersection is non- empty.

See also

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