دالة تباينية (2)

دالة تباينية ولكنها غير شمولية (ليست بدالة تقابلية)

دالة تباينية وشمولية في آن واحد (هي دالة تقابلية)

دالة غير تباينية ولكنها شمولية

Injective functions. Diagramatic interpretation in the Cartesian plane, defined by the mapping f : X →Y, where y = f(x), X = domain of function, Y = range of function, and im(f) denotes image of f. Every one xin X maps to exactly one unique y in Y. The circled parts of the axes represent domain and range sets – in accordance with the standard diagrams above.

Not an injective function. Here X₁ and X₂ are subsets of X, Y₁ andY₂ are subsets of Y: for two regions where the function is not injective because more than one domain element can map to a single range element. That is, it is possible for more than one x in Xto map to the same y in Y.

Making functions injective. The previous function f : X → Y can be reduced to one or more injective functions (say) f : X₁ → Y₁ and f : X₂ → Y₂, shown by solid curves (long-dash parts of initial curve are not mapped to anymore). Notice how the rule f has not changed – only the domain and range. X₁ and X₂ are subsets of X, Y₁ and Y₂ are subsets of R: 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 x in X maps to one y in Y.