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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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Discrete math questions

Looking for some answers to discrete math questions. Must show work so that I can understand how you achieved the results.

See attached.

1. Note the contrapositive of the definition of one-to-one function given on of the text is: If a ≠ b then f(a) ≠ f(b). As we know, the contrapositive is equivalent to (another way of saying) the definition of one-to-one.
a. Consider the following function f: R → R defined by f(x) = x2 - 9 . Use the contrapositive of the definition of one-to-one function to determine (no proof necessary) whether f is a one-to-one function. Explain

b. Compute f ° f.

c. Let g be the function g: R → R defined by g(x) = x3+ 3. Find g -1
Use the definition of g-1 to explain why your solution, g-1 is really the inverse of g.

2. Compute the double sums.

3. (See attached)

Compute:
(a) AC+ BC (It is much faster if you use the distributive law for matrices first.)

(b) 2A - 3A
(c) Perform the given operation for the following zero-one matrices.

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