Intersection

{{Unreferenced Section|date=June 2023}}

There can be more than one primitive object, such as points (pictured above), that form an intersection. The intersection can be viewed collectively as all of the shared objects (i.e., the intersection operation results in a set, possibly empty), or as several intersection objects (possibly zero).

File:Set intersection exemplified by road intersection.jpg

{{Main|Intersection (set theory)}}

The intersection of two sets {{math|A}} and {{math|B}} is the set of elements which are in both {{math|A}} and {{math|B}}. Formally,

:$A\; \backslash cap\; B\; =\; \backslash \{\; x:\; x\; \backslash in\; A\; \backslash text\{\; and\; \}\; x\; \backslash in\; B\backslash \}$.{{Cite book|url=https://books.google.com/books?id=LBvpfEMhurwC|title=Basic Set Theory|last1=Vereshchagin|first1=Nikolai Konstantinovich|last2=Shen|first2=Alexander|date=2002-01-01|publisher=American Mathematical Soc.|isbn=9780821827314|language=en}}

For example, if $A\; =\; \backslash \{1,\; 3,\; 5,\; 7\backslash \}$ and $B\; =\; \backslash \{1,\; 2,\; 4,\; 6\backslash \}$, then $A\; \backslash cap\; B\; =\; \backslash \{1\backslash \}$. A more elaborate example (involving infinite sets) is:

: $A\; =\; \backslash \{\; x:\backslash text\{\; x\; is\; an\; even\; integer\}\; \backslash \}$

: $B\; =\; \backslash \{\; x:\backslash text\{\; x\; is\; an\; integer\; divisible\; by\; 3\}\; \backslash \}$$\backslash text\{\; ,\; then\}$

: $A\; \backslash cap\; B\; =\; \backslash \{6,\; 12,\; 18,\; \backslash dots\backslash \}$

As another example, the number {{math|5}} is not contained in the intersection of the set of prime numbers {{math|{2, 3, 5, 7, 11, …} }} and the set of even numbers {{math|{2, 4, 6, 8, 10, …} }}, because although {{math|5}} is a prime number, it is not even. In fact, the number {{math|2}} is the only number in the intersection of these two sets. In this case, the intersection has mathematical meaning: the number {{math|2}} is the only even prime number.

{{excerpt|Intersection (geometry)}}

Intersection is denoted by the {{unichar|2229|intersection}} from Unicode Mathematical Operators.

The symbol {{unichar|2229}} was first used by Hermann Grassmann in Die Ausdehnungslehre von 1844 as general operation symbol, not specialized for intersection. From there, it was used by Giuseppe Peano (1858–1932) for intersection, in 1888 in Calcolo geometrico secondo l'Ausdehnungslehre di H. Grassmann.{{Cite book|url=https://books.google.com/books?id=5LJi3dxLzuwC|title=Calcolo geometrico secondo l'Ausdehnungslehre di H. Grassmann: preceduto dalle operazioni della logica deduttiva|last=Peano|first=Giuseppe|date=1888-01-01|publisher=Fratelli Bocca|language=it|location=Torino}}{{Cite book|url=https://books.google.com/books?id=bT5suOONXlgC|title=A History of Mathematical Notations |last=Cajori|first=Florian|date=2007-01-01|publisher=Cosimo, Inc.|language=en|location=Torino|isbn=9781602067141 }}

Peano also created the large symbols for general intersection and union of more than two classes in his 1908 book Formulario mathematico.{{Cite book|title=Formulario mathematico, tomo V|last=Peano|first=Giuseppe|date=1908-01-01|publisher=Edizione cremonese (Facsimile-Reprint at Rome, 1960)|language=it|page=82|oclc = 23485397|location=Torino}}{{URL|http://www.math.hawaii.edu/~tom/history/set.html|Earliest Uses of Symbols of Set Theory and Logic}}

• Constructive solid geometry, Boolean Intersection is one of the ways of combining 2D/3D shapes
• Dimensionally Extended 9-Intersection Model
• Meet (lattice theory)
• Intersection (set theory)
• Union (set theory)

• {{MathWorld|Intersection}}

zh-yue:交點

Category:Broad-concept articles