One-to-One (Injective) Function:
A function (f: A \rightarrow B) is said to be one-to-one (or injective) if each element in the domain (A) maps to a distinct element in the codomain (B).
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In other words, no two different elements in the domain map to the same element in the codomain.
- Mathematical Definition: For all (x_1, x_2 \in A), if (f(x_1) = f(x_2)), then (x_1 = x_2).
- Example: Consider the function (f: \mathbb{R} \rightarrow \mathbb{R}) defined by (f(x) = 2x). This function is one-to-one because if (f(x_1) = f(x_2)), then (2x_1 = 2x_2), and hence (x_1 = x_2).
Onto (Surjective) Function:
A function (f: A \rightarrow B) is said to be onto (or surjective) if every element in the codomain (B) is the image of at least one element in the domain (A). In other words, the range of the function covers the entire codomain.
- Mathematical Definition: For every (y \in B), there exists at least one (x \in A) such that (f(x) = y).
- Example: Consider the function (f: \mathbb{R} \rightarrow \mathbb{R}) defined by (f(x) = x^2). This function is not onto because there are elements in (\mathbb{R}) (negative values) that do not have pre-images in (\mathbb{R}) under this function.
Summary:
- One-to-One (Injective): No two different elements in the domain map to the same element in the codomain.
- Onto (Surjective): Every element in the codomain is covered by the range of the function.
A function can be both one-to-one and onto, in which case it is called a bijective function. Bijective functions have the property that each element in the domain uniquely corresponds to an element in the codomain, and every element in the codomain has a unique pre-image in the domain.