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Discrete Mathematical Definitions

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Could you give me a "working" definition of each term and an example of how they are used if possible.

Terms:
- Image
- Mapping
- Range
- Codomain
- Domain
- Surjective
- Injective
- Bijective
- One to one.

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1. Mapping
A mapping (also called a function) is a relation between a set of inputs X to a set of outputs Y. There is a restriction that each value of X gets mapped to only one value of Y.
We might write f: X -> Y to denote a mapping from X (the domain) to Y (the codomain).

For example, let X = {1, 2, 3} and Y={3, 6, 9}, and define a relation f(x) = 3*x. Then, f is a valid mapping. An equally valid mapping is defined by: h(1) = 6, h(2) = -1, h(3) = 10.

If instead we have a relation g: {1, 2, 3} -> {1, 4, 6, 7}, which simultaneously maps g(2) = 4 and g(2) = 6 (and keeps g(1) = 1, g(3) = 7), then g is not a valid mapping.

X and Y can also be continuous. For example, let X be the entire real line. Then let k(x) = 3*x, as before, in which case Y would also be the entire real line.

2. Image
The image of a mapping is the set of outputs Y that can be obtained from applying the mapping to the set of inputs X. In the above ...

Solution Summary

The solution describes discrete mathematical definitions.

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