1. Let n be an integer. Prove that the additive inverse (see Fact 3) of n is unique. (Side note: As a result, it makes sense to say "the" additive inverse. We should NOT use the word "the" unless it is unique. )
1. Let n be an integer. Prove that the additive inverse (see Fact 3) of n is unique. (Side note: As a result, it makes sense to say "the" additive inverse. We should NOT use the word "the" unless it is unique. )
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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For Questions 1,6,7,pleasae Thank you!
Transcribed Image Text:# Math 103: HW 3
For this homework, you can use the following facts **without citing**.
1. **Basic algebra**, such as \(2 + 2 = 4\), \(1 - 3 = -2\), \(2 \cdot 4 = 8\), \(0 \cdot x = 0\), \(1 \cdot x = x\), \(2 \cdot x = x + x\).
2. For all integers \(x, y, z\), we have the **Associative Law**: \((x + y) + z = x + (y + z)\) and \((xy)z = x(yz)\); thus we are allowed to write \(x + y + z\) and \(xyz\). We also have the **Commutative Law**: \(x + y = y + x\) and \(xy = yx\).
3. **Distributive law**: For all \(a, b, c \in \mathbb{C}\), we have \((a + b)(c + d) = ac + ad + bc + bd\). In particular, \((a + b)^2 = a^2 + 2ab + b^2\).
4. Let \(a, b, c\) be real numbers with \(b > c\). Then \(a + b > a + c\).
5. Let \(a, b, c\) be real numbers with \(a > 0\) and \(b > c\). Then \(ab > ac\).
Now here are some definitions / facts that you **need to cite** when used. The citing instructions are given after the list.
1. An integer \(n\) is **even** if there exists an integer \(k\) such that \(n = 2k\).
2. An integer \(n\) is **odd** if there exists an integer \(k\) such that \(n = 2k + 1\).
3. All integers are even or odd.
4. If \(a, b \in \mathbb{C}\) and \(a, b \neq 0\), then \(ab \neq 0\). In particular, if \(a^2 = 0\), then \(a = 0\).
5. A real number \(x
![### Linear Algebra and Analysis Concepts
#### 2. Determinants and Matrices
Recall that the determinant is a function from the set of 2 × 2 matrices over ℝ to the set of real numbers:
\[
\text{det} : \left\{ \left( \begin{array}{cc} a & b \\ c & d \end{array} \right) \mid a, b, c, d \in \mathbb{R} \right\} \to \mathbb{R}
\]
The determinant of a 2 × 2 matrix is calculated as follows:
\[
\text{det} \left( \begin{array}{cc} a & b \\ c & d \end{array} \right) = ad - bc
\]
**(a)** Prove that for all real numbers \( x \), there exists a 2 × 2 matrix over ℝ such that its determinant is \( x \).
**(b)** Is the matrix in part (a) unique? Provide a proof or a counterexample.
#### 3. Absolute Value Function
We define the absolute value function as follows: for all \( x \in \mathbb{R} \), we have
\[
|x| := \begin{cases}
x & \text{if } x \geq 0 \\
-x & \text{if } x < 0
\end{cases}
\]
Prove that for all \( x \in \mathbb{R} \), we have \( x^2 \geq 0 \).
#### 4-7. Additional Mathematical Proofs
4. Let \( x \) be a real number. Prove that if \( x^2 = x \), then \( x < 2 \).
5. Prove that if \( n \) is an integer, then \( n^2 + 3n + 1 \) is odd.
6. Let \( a, b \) be integers. Prove that if \( a + b \) is even, then \( a - b \) is even.
7. Let \( a, b \) be integers. Prove that if \( ab \) is even, then \( a \) or \( b \) is even.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fed3eef2a-70e8-485e-b91d-563de65921b3%2F0bc580b8-00c5-4576-ba83-1081f54086f6%2Fa8gm92f_processed.png&w=3840&q=75)
Transcribed Image Text:### Linear Algebra and Analysis Concepts
#### 2. Determinants and Matrices
Recall that the determinant is a function from the set of 2 × 2 matrices over ℝ to the set of real numbers:
\[
\text{det} : \left\{ \left( \begin{array}{cc} a & b \\ c & d \end{array} \right) \mid a, b, c, d \in \mathbb{R} \right\} \to \mathbb{R}
\]
The determinant of a 2 × 2 matrix is calculated as follows:
\[
\text{det} \left( \begin{array}{cc} a & b \\ c & d \end{array} \right) = ad - bc
\]
**(a)** Prove that for all real numbers \( x \), there exists a 2 × 2 matrix over ℝ such that its determinant is \( x \).
**(b)** Is the matrix in part (a) unique? Provide a proof or a counterexample.
#### 3. Absolute Value Function
We define the absolute value function as follows: for all \( x \in \mathbb{R} \), we have
\[
|x| := \begin{cases}
x & \text{if } x \geq 0 \\
-x & \text{if } x < 0
\end{cases}
\]
Prove that for all \( x \in \mathbb{R} \), we have \( x^2 \geq 0 \).
#### 4-7. Additional Mathematical Proofs
4. Let \( x \) be a real number. Prove that if \( x^2 = x \), then \( x < 2 \).
5. Prove that if \( n \) is an integer, then \( n^2 + 3n + 1 \) is odd.
6. Let \( a, b \) be integers. Prove that if \( a + b \) is even, then \( a - b \) is even.
7. Let \( a, b \) be integers. Prove that if \( ab \) is even, then \( a \) or \( b \) is even.
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