03-05-03

# 03-05-03 - Lecture 14 Andrei Antonenko 1 Functions In...

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Unformatted text preview: Lecture 14 Andrei Antonenko March 05, 2003 1 Functions In previous lectures we worked with algebraic structures — sets with operations defined on them. Now we will consider another important thing in mathematics — functions. Let A and B be 2 sets. Function f from A to B can be considered as a rule, which allows us to get an element from B for any element from A . The notation for a function from the set A to the set B is: f : A → B . Set A is called the domain of a function f . We will often use the following notation: x 7→ f ( x ), which denotes that x maps to f ( x ), i.e. applying f to x we get f ( x ). Now let’s consider any element x from A . Then f ( x ) ∈ B is called the image of x . Moreover we can consider the subset A ⊂ A . Then by f ( A ) we will denote the set which contains images of all the elements from A and it will be called the image of A . Let’s consider any subset in B , say, B ∈ B . Then by f- 1 ( B ) we will denote all elements from A , whose images are in B . f- 1 ( B ) will be called the inverse image of preimage of B . Example 1.1. Consider the function f ( x ) = x 2 . This function is defined for any real number, and maps them to nonnegative real numbers. If R + denotes positive numbers, then f : R → R + ∪ { } , where R + ∪ { } is the set of all nonnegative numbers....
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03-05-03 - Lecture 14 Andrei Antonenko 1 Functions In...

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