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Mapping
Let A and B be two non - empty sets. A mapping f from A to B is a rule, that assigns to each element x of a definite element y in B.
One-to-one or injective mapping
A mapping f: A ↦B is said to be injective (or one-to-one) if, for each pair of distinct elements of A, their f-images are distinct.
Bijective mapping
A mapping f: A →B is said to be bijective if f is both injective and surjective. Thus f: A↦B is injective if x’ ≠x”, in A implies f(x’)≠ f(x”) in B. In this case, each element of B has at most one pre-image.
If f is surjective, each element of B has at least one pre-image. If f is bijective, each element of B has exactly one pre-image. An injective mapping is called an injection, a surjective mapping a surjection and a bijective mapping a bijection.
Constant mapping
A mapping f: A↦ B is said to be a constant mapping (a constant function) if f maps each element of A to one and the same element of B, i.e. f(A) is a singleton set.
For example, the mapping f: R→R defined by f(x) = 2, x∈ R, is a constant mapping. Here f(R)={2}.
Identity mapping
A mapping f: A↦ A is said to be the identity mapping on A if f (x)=x3, x∈A. and denoted by iA
inverse mapping
Let f: AB is a mapping. If there exists a mapping g: B↦ A such that gof = iA then g is said to be a left inverse of f. If there exists a mapping g: B↦ A such that fog =iB then g is said to be a right inverse of f.
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