Surjective proof example
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How to prove a function is surjective and injective!
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6.3: Injections, Surjections, and Bijections
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Functions are frequently used in mathematics to define and describe certain relationships between sets and other mathematical objects.
In addition, functions can be used to impose certain mathematical structures on sets. In this section, we will study special types of functions that are used to describe these relationships that are called injections and surjections.
Before defining these types of functions, we will revisit what the definition of a function tells us and explore certain functions with finite domains.
Preview Activity \(\PageIndex{1}\): Functions with Finite Domains
Let \(A\) and \(B\) be sets.
Given a function \(f : A \to B\), we know the following:
- For every \(x \in A\), \(f(x) \in B\). That is, every element of \(A\) is an input for the function \(f\). This could also be stated as follows: For each \(x \in A\), there exists a \(y \in B\) such that \(y = f(x)\).
- For a given \(x \in A\), there is exactly one \(y \in B\) such that \(y = f(x)\).
The definition of a function doe
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