January:16(Fri)

MTH 225(M) Exercise-Set #2

Due January 23 (Friday) at Lecture

Read our textbook's Chapter 1.

Please do all of these exercises in teams of (at most) two. Such teamwork earns a 5% bonus for you.

Show intermediate work/explanations for obtaining answers.

[6 points] Do Exercise 1.1.24 of Section 1.1 of our textbook, this time following its instructions. And actually for b and c, copy here the solutions from the previous exercise regarding this material.
Do the following exercises of Section 1.3 of our textbook:
- [2 points] 1.3.3 Explain the answers (which are in the back of the book).
- [4 points] 1.3.8
- [5 points] 1.3.10
- [7 points] 1.3.12 For d-g, use the rules presented in lectures for determining whether quantified formulas are true or false.
- [3 points] 1.3.36
- [4 points] 1.3.52
Show intermediate work/explanations for obtaining answers.
Do the following exercises of Section 1.4 of our textbook:
- [6 points] 1.4.4
- [6 points] 1.4.6
- [6 points] 1.4.8
  For b, actually use the following version of the sentence:
  "No student at your school has ever been a contenstant on any television quiz show.
  And for e, actually use the following version of the sentence:
  At least two different students from your school have been contestants on Jeopardy.
  For models for your answers, look at the solutions in the back of the book for 1.4.9 etc.
  And think carefully about how you can express the condition that two students are different — how do you express the condition in Java that the values of two variables say x1 and x2 are different?
- [4 points] 1.4.20
- [11 points] 1.4.27a,c,d,e,g
  Do these using the rules presented in lectures for determining whether quantified formulas are true or false.
  Note the solutions in the back of the book.
- [8 points] 1.4.28a,b,c,f
  Do these using the rules presented in lectures for determining whether quantified formulas are true or false.
- [5 points] 1.4.30
Show intermediate work/explanations for obtaining answers.
Do the following exercises of Section 1.2 of our textbook (one for the second time) like the back-of-the-book solutions for 1.2.11 and 1.2.17-27 (odd numbers):
- [2 points] 1.2.12a
- [3 points] 1.2.24
[15 points] Using simplifications specified in my Lecture-Module on "Basics of Symbolic Proofs", simplify the following logic formulas as much as possible:
1. ```
    ¬[true → ¬P]
```
2. ```
    P  ∨  [true → (true ∨ Q)]
```
3. ```
    ¬(P ∨ P) →  ¬Q ∨ false
```
4. ```
    ([P ∧ P]   ∧   ¬[¬(Q ∨ false)])
        ∨
    ¬([true → P]   ∧   [(R ∨ ¬R) → Q])
```
In each case, as shown in lecture(s), show the steps of simplication — including telling the names of the simplications that you use, and showing how the formula looks after each individual simplification.
Acknowledgement: This exercise was derived from "The Deductive Foundations of Computer Programming" by Manna & Waldinger.
Do the following exercises of Section 1.2 of our textbook (one for the second time and one for the third time) like Examples 6-8 presented on pages 26-27:
- [4 points] 1.2.12a (particularly like Example 8)
- [3 points] 1.2.22 (particularly like Examples 6-7)
- [3 points] 1.2.24 (particularly like Examples 6-7)
Show intermediate work/explanations for obtaining answers.