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An '''inverse image''' in ] is defined in the following way: An '''inverse image''' in ] is defined in the following way:


Consider a ] f which maps from some ] A to some ] B. Consider a ] f which maps from some ] A to some set B.
Let C be a ] of B, then the ''inverse image'' of C under the ] f, written f<sup>&minus;1</sup>(C), is defined as the ] of all ] in A which map into C. Let C be a ] of B, then the ''inverse image'' of C under the function f, written f<sup>&minus;1</sup>(C), is defined as the set of all ] in A which f maps into C.


Example: Example:


Take as ] A and B the ] 1, 2, 3,... Take as function f the ] that assigns to every ] its ]. Take as ] C of B all ] less than 10. Then the ''inverse image'' of C consists of the ] 1, 2 and 3, since only these ] have a ] less than 10, namely 1, 4 and 9. Take as sets A and B the ]: 1, 2, 3, … Take as function f the ] that assigns to every number its ]. Take as subset C of B all natural numbers less than 10. Then the ''inverse image'' of C consists of the numbers 1, 2 and 3, since only these numbers have a square less than 10, namely 1, 4 and 9.

Revision as of 04:04, 20 July 2004

An inverse image in mathematics is defined in the following way:

Consider a function f which maps from some set A to some set B. Let C be a subset of B, then the inverse image of C under the function f, written f(C), is defined as the set of all elements in A which f maps into C.

Example:

Take as sets A and B the natural numbers: 1, 2, 3, … Take as function f the quadratic function that assigns to every number its square. Take as subset C of B all natural numbers less than 10. Then the inverse image of C consists of the numbers 1, 2 and 3, since only these numbers have a square less than 10, namely 1, 4 and 9.