Trying to understand proofs, help me solve this one

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In summary, to prove that f \circ g is also surjective, we need to show that for every z in the codomain of f \circ g, there exists an x in the domain of g such that f(g(x)) = z. This can be done by considering the functions f and g separately and showing that they are both onto functions.
  • #1
Suppose that [itex]f, g : \mathbb{R} \rightarrow \mathbb{R}[/itex] are surjective (ie onto functions with domain [itex]\mathbb{R}[/itex] and allowable output values [itex]\mathbb{R}[/itex]). Prove that [itex]f \circ g[/itex] is also surjective (ie, prove [itex]f \circ g[/itex] is also onto).

First of all, I have absolutely no math theory experience, so I don't really understand what's being asked for here.

I know that ℝ is the set of all real numbers, but I'm not sure what ℝ → ℝ represents.

Can someone explain to me the mathematical terms and give me a breakdown of how this problem would be solved?
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  • #2
f being an onto function means that for every real number y there exists x such that f(x) =y.
  • #3
[itex]f \colon A \to B[/itex] means a function with name [itex]f[/itex] from the domain [itex]A[/itex] into the codomain [itex]B[/itex]. For instance, consider the function [itex]f\colon \mathbb{R} \to \mathbb{R}\times \mathbb{R}[/itex] defined by the rule [itex]f(x) = (x,x)[/itex]. In this example the domain (e.g., the set [itex]x[/itex] is in) is the real line [itex]\mathbb{R}[/itex] and the codomain is the set [itex]\mathbb{R}\times \mathbb{R}[/itex], e.g., the 2D plane. This function is not onto though, because there are points [itex](a,b) \in \mathbb{R}\times \mathbb{R}[/itex] such that there is no [itex]x \in \mathbb{R}[/itex] with [itex]f(x) = (a,b)[/itex]. To prove this, let [itex]a, b[/itex] be real with [itex]a \neq b[/itex] and assume [itex]f(x) = (a,b)[/itex]. Since [itex]f(x) = (x,x) = (a,b)[/itex] must hold, then [itex]x = a[/itex] and [itex]x = b[/itex] must also be true, but [itex]a \neq b[/itex] so it cannot. The only parts of the codomain that are hit are those on the line [itex]y = x[/itex] which can be written as [itex]L = \{(a,a)\colon a \in \mathbb{R}\}[/itex]. In this case [itex]L[/itex] is the range of [itex]f[/itex] (it's also called the image of [itex]f[/itex]), which is always a subset of the codomain.

However, the function [itex]g\colon \mathbb{R} \to L[/itex] defined by the rule [itex]g(x) = f(x) = (x,x)[/itex] is onto. To prove this, let [itex](a,a) \in L[/itex]. Then we see [itex]a \in \mathbb{R}[/itex] and [itex](a,a) = g(a)[/itex]. Since the element [itex](a,a)[/itex] chosen from [itex]L[/itex] was arbitrary, we see that [itex]g\colon \mathbb{R} \to L[/itex] is an onto function. In this case the range and the codomain of the function [itex]g[/itex] are the same: namely, [itex]L[/itex].

This is really all you need to do in your problem: pick an arbitrary point [itex]z[/itex] in the codomain of [itex]f \circ g[/itex] and find a [itex]y[/itex] in the domain of [itex]f[/itex] such that [itex]f(y) = z[/itex]. Then find an [itex]x[/itex] in the domain of [itex]g[/itex] such that [itex]g(x) = y[/itex]. E.g., you find an [itex]x[/itex] such that [itex]g(x) = y[/itex] and [itex]f(y) = z[/itex] so that [itex]f(g(x)) = z[/itex].
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1. What are proofs and why are they important in science?

Proofs are a logical and systematic way of demonstrating the truth or validity of a statement or theorem. In science, proofs are crucial because they provide evidence and support for theories and hypotheses, allowing us to make informed conclusions about the natural world.

2. How do I approach solving a proof?

To solve a proof, it is important to carefully read and understand the given statements and definitions. Then, use logical reasoning and known mathematical principles to make deductions and reach a conclusion. It is also helpful to break the proof into smaller steps and to try different approaches if one method is not working.

3. What are some common mistakes to avoid when solving a proof?

Some common mistakes to avoid when solving a proof include assuming what you are trying to prove, using incorrect or incomplete definitions, and skipping steps without proper justification. It is important to be precise and thorough in your reasoning and to double-check your work for any errors.

4. How can I improve my ability to understand and solve proofs?

Practice and patience are key in improving your ability to understand and solve proofs. It is also helpful to seek out resources such as textbooks, online tutorials, and practice problems to gain a better understanding of different proof techniques and strategies.

5. Can proofs be solved using different methods?

Yes, there are often multiple ways to approach and solve a proof. It is important to use the method that makes the most sense to you and that you are most comfortable with. However, it is also beneficial to try different methods as it can help improve your problem-solving skills and deepen your understanding of the concept being proved.

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