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m Reversed the order. Clearer and exactly parallels the statement in English. (≠ is math for “distinct”) Tags: Visual edit Mobile edit Mobile web edit Advanced mobile edit |
Sadnoumena (talk | contribs) m →Gallery: Changed order of first two images of graphs. More coherent now with their descriptions. |
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|Image:Non-injective function2.svg|Making functions injective. The previous function <math>f : X \to Y</math> can be reduced to one or more injective functions (say) <math>f : X_1 \to Y_1</math> and <math>f : X_2 \to Y_2,</math> shown by solid curves (long-dash parts of initial curve are not mapped to anymore). Notice how the rule <math>f</math> has not changed – only the ___domain and range. <math>X_1</math> and <math>X_2</math> are subsets of <math>X, Y_1</math> and <math>Y_2</math> are subsets of <math>Y</math>: for two regions where the initial function can be made injective so that one ___domain element can map to a single range element. That is, only one <math>x</math> in <math>X</math> maps to one <math>y</math> in <math>Y.</math>▼
|Image:Non-injective function1.svg|Not an injective function. Here <math>X_1</math> and <math>X_2</math> are subsets of <math>X, Y_1</math> and <math>Y_2</math> are subsets of <math>Y</math>: for two regions where the function is not injective because more than one ___domain [[Element (mathematics)|element]] can map to a single range element. That is, it is possible for {{em|more than one}} <math>x</math> in <math>X</math> to map to the {{em|same}} <math>y</math> in <math>Y.</math>
▲|Image:Non-injective function2.svg|Making functions injective. The previous function <math>f : X \to Y</math> can be reduced to one or more injective functions (say) <math>f : X_1 \to Y_1</math> and <math>f : X_2 \to Y_2,</math> shown by solid curves (long-dash parts of initial curve are not mapped to anymore). Notice how the rule <math>f</math> has not changed – only the ___domain and range. <math>X_1</math> and <math>X_2</math> are subsets of <math>X, Y_1</math> and <math>Y_2</math> are subsets of <math>Y</math>: for two regions where the initial function can be made injective so that one ___domain element can map to a single range element. That is, only one <math>x</math> in <math>X</math> maps to one <math>y</math> in <math>Y.</math>
|Image:Injective function.svg|Injective functions. Diagramatic interpretation in the [[Cartesian plane]], defined by the [[Map (mathematics)|mapping]] <math>f : X \to Y,</math> where <math>y = f(x),</math> {{nowrap|<math>X =</math> ___domain of function}}, {{nowrap|<math>Y = </math> [[range of a function|range of function]]}}, and <math>\operatorname{im}(f)</math> denotes image of <math>f.</math> Every one <math>x</math> in <math>X</math> maps to exactly one unique <math>y</math> in <math>Y.</math> The circled parts of the axes represent ___domain and range sets— in accordance with the standard diagrams above
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