All Homework

Homework 1

(8) AIDM Problem 1.8 (page 31)
Hint: Use a truth table and a few words for one of the proofs. The other proof might start with something like "The proposition is only false when..."
(16) AIDM Problem 1.12 (page 32)
Hint: Make sure for parts (c) and (d) you only use ↓. You are not allowed to use ¬ (the most common mistake students seem to make). You might want to start by figuring out how to implement ¬ using ↓.

(5) AIDM Problem 1.15e. Show every step!
(5) AIDM Problem 1.16f.
(5) AIDM Problem 1.17e. Justify your answer by making it clear that you understand what it is saying and then why it is true or false.
(5) AIDM Problem 1.18f.

(6) Convert \((p\oplus q)\wedge(q\vee r)\) to disjunctive normal form using procedure 1.93 from the textbook. Make sure you show all of the steps!
(6) Use the definition of odd and even to prove that if \(n\) is odd, then \(n^2+n\) is even.

(6) AIDM Problem 2.4 (page 71). To be clear, you are not being asked to prove anything in this problem. You are asked to rephrase the given statement and justify the rephrasing.
(6) AIDM Problem 2.19 (page 72)

(6) AIDM Problem 2.10 (page 71). Make sure your proof is based on the definition of even.
(4) AIDM Problem 2.22. (Easy! Don't overthink it.)
(6) AIDM Problem 2.23. (Also pretty easy if you realize that \(1=1^2\).)

(4) AIDM Problem 3.1 on page 129.
(8) Let \(U=\{A,B,C,...,Z\}\) (the capital letters of the English alphabet) be the universal set and \(S=\{A, B, C, D\}\).
1. What is \(|S|\)?
2. What is \(|P(S)|\)?
3. How many subsets does \(S\) have?
4. What is \(|\overline{S}|\)
(2) Is \(\mathbb{Z}^+\cup\mathbb{Z}^-=\mathbb{Z}\)? Clearly explain.
(8) Let \(A=\{2x | x\in\mathbb{Z}\}\) and \(B=\{3x | x\in\mathbb{Z}\}\). Express each of the following using set notation.
1. \(A\cup B\)
2. \(A\cap B\)
3. \(A\setminus B\)
4. \(A\times B\)

(8) AIDMA Problem 3.3b. (Just part b!)
Hint: Since you are proving equality, this will involve two proofs—one for set containment in each direction.
(4) AIDM Problem 3.6
(4) AIDM Problem 3.8ef
(4) AIDM Problem 3.9ef
Bonus! (4) AIDM Problem 3.11. (FYI, I am fairly strict on grading bonus problems.)

(6) Let \(f:\mathbb{Z}\rightarrow\mathbb{Z}\) be defined by \(f(x)=x^3-x\). Answer each of the following, with a brief justification.
1. Is \(f\) one-to-one?
2. Is \(f\) onto?
3. Is \(f\) invertible?
(8) Let \(f\) be a function that maps a person on earth to the number of pets that they have owned.
1. What is the domain of \(f\)?
2. What is the codomain of \(f\)?
3. The exact range of \(f\) is difficult to know for sure. But is it the same as the codomain? Explain.
4. Is the range of \(f\) a subset of the codomain? Explain.
(6) Let \(f(x)=e^x\) and \(g(x)=3x^2-4x+7\)

(8) Let \(f:\mathbb{R}\rightarrow\mathbb{R}\) be defined by \(f(x)=x^3-1\).
1. Prove that \(f\) is one-to-one.
2. Prove that \(f\) is onto.
3. Prove that \(f\) is invertible.
4. Find \(f^{-1}(x)\).
(8) AIDM Problem 3.24 (Page 131).
You have to prove 3 things! One of them is so easy it can seem confusing. The other two are not that difficult. Look at the examples in the book for inspiration.

(4) AIDM Problem 4.2 (page 179)
(6) AIDM Problem 4.5 (page 179).
Make sure you show the work that helped you figure out the formula!

(5) AIDM Problem 4.25 (Page 181). Make sure you show all of the steps! This problem is pretty straightforward—Just plug \(A\) in and evaluate the lefthand side until it simplifies to \(\textbf{0}_2\).
(5) Let \(M = \begin{bmatrix} 0 & 1 \cr 1 & 1 \cr\end{bmatrix}\). Compute \(M^{12}\). Show all of your work.
(Hint: If you are clever, you can do this with 4 matrix multiplications instead of 11.)

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