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Chapter 6: Problem 13

Prove that the composition of onto functions is onto.

Short Answer

Expert verified
The composition of onto functions is onto. This was proven by showing that for every element \(z\) in the range of the composition, there exists an element \(x\) in the domain such that when \(x\) is taken through the function composition, \(z\) is obtained.

Step by step solution

01

Understand Function Composition

Let's denote two onto functions as \(f: X \rightarrow Y\) and \(g: Y \rightarrow Z\). The composition of these two functions is a function \(h = g \circ f: X \rightarrow Z\), which can be described as \(h(x) = g(f(x))\) for all \(x\) in \(X\).
02

Prove the Composition is Onto

To prove the composition \(h\) is onto, it needs to be shown that for every \(z\) in \(Z\), there exists an \(x\) in \(X\) such that \(h(x) = z\). Since \(g\) is onto, for each \(z\) in \(Z\), there exists an \(y\) in \(Y\) such that \(g(y) = z\). Likewise, as \(f\) is onto, for the chosen \(y\) in \(Y\), there exists an \(x\) in \(X\) such that \(f(x) = y\). Therefore, we have \(h(x) = g(f(x)) = g(y) = z\). So, for every \(z\) in \(Z\), there exists an \(x\) in \(X\) such that \(h(x) = z\), which is what it means for \(h\) to be onto.
03

Conclusion

Based on the above, it's proven that the composition \(h = g \circ f\) of two onto functions \(f\) and \(g\) is onto. This concludes the proof.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Onto Function
An onto function, also referred to as a surjective function, is a type of function with a particular property. It ensures that every element in the codomain is mapped to by at least one element from the domain. In other words, for every element \( z \) in the codomain \( Z \), there exists an element \( x \) in the domain \( X \) such that \( f(x) = z \). This means that the entire range or output of the function is covered.
This concept is very important in mathematics, especially in proving properties about function compositions. An onto function ensures no gap in target coverage, making it a crucial piece in seamlessly linking one function to another through composition.
Often, the proof involving onto functions will follow a pattern where you demonstrate for an arbitrary element in the target set that there is a corresponding element in the source set.
  • Ensures complete coverage of the codomain.
  • Every output value has a pre-image.
  • Verification typically involves testing arbitrary examples or establishing a pattern.
Function Mapping
Function mapping describes how functions relate elements from one set to another. In mathematical terms, for a function \( f: X \to Y \), every element \( x \) in the domain \( X \) is paired with a unique element \( y \) in the codomain \( Y \).
Function mappings are crucial in understanding how functions operate and define the nature of the relationship between input and output values.
In the world of functions, there are different types such as one-to-one (injective), onto (surjective), and bijective functions. Each type defines a unique mapping behavior. However, onto functions (as we discussed earlier) are essential when talking about composition.
  • A function \( f: X \to Y \) associates each \( x \) in \( X \) with \( f(x) \) in \( Y \).
  • A composition of functions adds layers of mappings transforming an initial input through successive functions.
  • The nature of mapping (onto, one-to-one) defines how tightly elements are paired.
Understanding mapping is key when working with the composition of functions, especially when you want to prove that a result (like an onto function) holds under composition.
Proof in Mathematics
Proofs in mathematics are logical arguments that verify the truth of a statement based on axioms, definitions, and previously established theorems. When constructing a proof, it's important to clearly define the problem and methodically work through logical steps to reach a conclusion.
For example, in proving that a composition of two onto functions is onto, we start by analyzing both functions separately. We need to show that for every result in the final set, there is at least one starting point mapped through both functions to reach it.
Mathematical proofs often involve:
  • Defining the given problem clearly, including all conditions and properties.
  • Breaking the problem into smaller, manageable steps.
  • Employing logical reasoning to connect these steps and arrive at a conclusion.
Proofs are vital tools in mathematics, providing assurance that certain truths hold universally under defined conditions. This process not only validates mathematical statements but also deepens understanding by dissecting complex problems into more comprehensible parts.

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Most popular questions from this chapter

Find the kernel and image of the zero function \(Z: V \rightarrow W\) defined by \(Z(v)=0_{W}\) for all \(v \in V\).

Explain why the exponential function exp : \(\mathbb{R} \rightarrow \mathbb{R}\) does not have an inverse. What simple modification will remedy this problem?

Consider the function \(R_{\varphi}: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}\) that rotates each point about the \(x\)-axis through an angle of \(\varphi\). a. Give a geometric argument that \(R_{s}\) is linear. b. Find a matrix \(A_{\varphi}\) such that \(R_{\psi}(\mathrm{v})=A_{\psi} \mathrm{v}\) for all \(\mathrm{v} \in \mathbb{R}^{3}\). c. Set up appropriate notation and derive similar results for rotations about the \(y\)-axis and rotations about the z-axis.

a. Describe a one-to-one function from the natural numbers onto the even natural numbers. b. Describe a one-to-one function from the natural numbers onto the integers. c. Describe a one-to-one function from the natural numbers onto the rational numbers. d. Describe a one-to-one function from the interval \([0,1)\) onto the interval \([0,1]\). e. Describe a function from \(\mathbb{R}\) onto \(\mathbb{R}^{2}\). Is it possible for such a function to be one-to-one?

Let \(R_{\theta}: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}\) be the linear function that rotates each point counterclockwise about the origin through an angle of \(\theta\). a. For any value of \(\theta\), give geometric arguments to show that \(R_{\theta}\) is one-to-one and onto. b. Explain geometrically why \(R_{\theta} \circ R_{\varphi}=R_{\theta+\varphi}\). c. Use the result of part b to show that \(R_{\theta}^{-1}=R_{-\theta}\). d. The matrix of a rotation relative to the standard basis for \(\mathbb{R}^{2}\) is given in the second example of Section 6.3. Verify directly that the product of the matrices of \(R_{\theta}\) and \(R_{\varphi}\) is the matrix of \(R_{\theta+\varphi}\). e. Verify directly that the matrix of \(R_{-\theta}\) is the inverse of the matrix of \(R_{\theta}\).
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