Math 260 Assignments May 19 to 23
Homework due Wednesday, May 21
Handout (4.1/3, 4.2/1, 2, 10h)
Hint for 10h: "x ∈∅" ≡ "FALSE" ≡ "Contradiction".
Logic Laws are on pages 17 & 19
Comment: Look at the Logic Implication Laws on page 17 for the last two Set Laws
given in 4.2/1
Monday, May 19
Sets, Subsets and set operations.
Read §4.1
Tuesday, May 20
Proofs with set operations.
DeMorgan’s Laws for Sets I
Read §4.2
Wednesday, May 21
More proofs with sets
Thursday, May 22
Big Unions and Big Intersections
Read §4.3
Friday, May 23
LAST Quiz & Worksheet
Quiz:
1. Set operations with intervals (not “Big” ones.)
2. Prove (A  B)c = A c  B c using a sequence of logically equivalent compound forms.
Group Work:
One of 4.2/14, 15, 17, 19, 20 or 21
Math 260 Assignments May 26 to June 6
Homework due Thursday, May 29
6.2/4, 14
Monday, May 26
Memorial Day – No School
Tuesday, May 27 (Lila Harper’s 59th birthday)
Introduction to PMI
Practice Problems (do not turn in): 6.1/4
Wednesday, May 28
Practice with PMI
Thursday, May 29
Getting ready for the Binomial Theorem (part I)
Friday, May 30
In Class PMI Worksheet. Do not turn in.
Monday, June 2
Getting ready for the Binomial Theorem (part II)
Practice Problems for Exam III handed out.
Tuesday, June 3
PMI proof of the Single Variable Binomial Theorem (B.T.)
Proof of the Two Variable B.T. using the Single Variable B.T.
N.B., You will be asked, on the Final Exam, to prove the Single Variable B.T., using PMI
on the Final Exam. (Not on Exam III).
Wednesday, June 4
Review for Exam III
Come prepared with questions!
Thursday, June 5
Exam III
Friday, March 14
Exam III returned. Discuss Final Exam.
Math 260 Assignments May 9 to May 16
Weekend reading assignment: Read §3.5
Homework due Monday, May 12
3.5/1, 11
Homework due Wednesday, May 14
3.7/5, 10(ii)
Monday, May 12
Reductio Ad Absurdum: Proof by Contradiction
1) Euclid’s proof that there are infinitely many primes.
Read §3.6 and §3.7
2)
2 is irrational.
Tuesday, May 13
Logs and contradiction.
Read §3.8
Handout on §3.8: Critique and score the proofs in the handout.
Wednesday, May 14
Grade and score the “proofs” in the handout.
Thursday, May 15
Review for Exam II.
Come prepared with questions!
Friday, May 16
Exam II
Overview of Exam II
Sixty points and seven “problems.”
1. a) Find the Logical Form for a given conditional.
b) Negate a logical form.
2. True/False similar to exercise 3.5/1
3. Find and simplify the contrapositive of a (quantified) conditional.
N.B., A conditional may have another conditional imbedded as either the antecedent or as
the consequent. That is, a conditional may have one of the forms:
(i) ( p → q) → r or (ii) p → (q → r).
It should be noted that these compound forms are NOT logically equivalent.
4. Given a divisibility test express it arithmetically (including quantifiers) and then apply
it to a pair of (base 10) positive integers.
5. Prove one direction of a divisibility test.
6. A mod 3 cases proof.
7. A contradiction proof: log b r is irrational for given r and b.
Practice Problem: Prove that log12 is irrational.
Homework due Tuesday, May 6
3.3/18
Homework due Wednesday, May 7
Handout: 3.4/1, 6
Extra Credit due Friday, February 14
3.3/21
Monday, May 5
The 7 and 13 divisibility tests
Tuesday, May 6
The Contrapositive Proof
Read §3.4
Wednesday, May 7
Go over HW 3.4/1
The Fundamental Theorem of Arithmetic (FTA)
Read §3.5
Thursday, May 8
Proof by cases
Friday, May 9
Quiz and Group Worksheet
Quiz
1. Find and simplify the contrapositive of a given conditional.
2. Given a divisibility test, express it arithmetically, including quantifiers.
3. Apply this divisibility test to a pair of given positive integers.
Group Work
1. Prove one direction of a divisibility test.
2. Do an even/odd contrapositive proof.
Practice Problems for the Group Work:
1. Prove for integers q and r, that if 19 (10q + r) , then 19 (q + 2r) .
2. Prove, for all integers x and y, that if 7x 2 − y 2 is odd, then x − y is odd.
Homework due Monday, May 12
3.5/1,11
Math 260 Assignments April 28 to May 2
Homework due Wednesday, April 30
3.1/1, 3, 4, 14
Monday, April 28
Introduction to the Direct Proof: The sum of two even numbers is even.
Read §3.1
Tuesday, April 29
More on the Direct Proof.
Wednesday, April 30
a b and The Division Algorithm, a mod d
Read §3.2 and §3.3
Thursday, May 1
Working with mod, A Few Divisibility Tests, The 9 Remainder Rule.
Friday, May 2
Quiz III and Group Worksheet #3
Quiz
(i) Use 9 Remainder Rule
(ii) Negate a conditional
(iii) Compute a mod d for given a and d.
Worksheet
Two proofs:
(i) Use the definitions of even and odd.
(ii) Use the definitions of x mod d and “divides”.
Math 260 Assignments April 21 to April 25
Monday, April 21
Go over Homework
Tuesday, April 22
The Distributive & Factor Laws for the quantifiers over “and” and “or”
Read §2.7
Wednesday, April 23
Review for Exam I – Come prepared with questions!
Thursday, April 24
Exam I
Friday, April 25
No class (I will be grading exams)
Review for Exam I
Sixty points, seven “problems”.
1. Use a truth table to prove that two compound forms are logically equivalent.
Recall that we do not use either of the symbols “ ≡ ” or “ ⇔ ” in a truth table.
2. True/False with quantifiers.
3. Negations: DeMoran’s Laws, negating quantifiers, negating conditionals.
4. Counterexamples: When is an object a counterexample? (e.g., see exercise 2.4/5)
Construct counterexamples.
5. Logical Forms, implicit quantifiers.
6. Rules of Inference: Modus Ponens, Modus Tollens, Converse Error, Inverse Error.
7. (True/False) The Distributive and Factor Laws for the quantifiers over ∧ and ∨ .
Math 260 Assignments April 11 to April 18
Homework due Monday, April 14
2.4/1, 2, 3, 4, 5 (Handout)
Extra Credit (due Friday, April 18): 2.4/6
Friday, April 11
Implicit Quantifiers
Read §2.4
Monday, April 14
Go over HW
Tuesday, April 15 (Euler’s Birthday)
Syllogisms and Rules of Inference
Read §2.5
Wednesday, April 16
Quiz II and Worksheet 2
Quiz
Implicit Quantifiers (see exercise 2.4/4)
Worksheet
True/False with quantifiers
Rules of Inference: Modus Ponens, Modus Tollens, Converse Error, Inverse Error
Thursday, April 17
Negations: DeMorgan, Conditional Negation Law, DeMorgan’s for Quantifiers
Read §2.6
Friday, April 18
Negating multiple quantifiers
Homework due Monday, April 21
Handout
Math 260 Assignments April 2 to April 10
Homework due Friday, April 4
2.1/1, 2d, 2f, 2h, 5
Comment: Do not turn in the truth tables for exercise 5.
Use a ruler to draw your truth tables.
Do your HW on one – side only! Otherwise, it will not be graded.
When you turn in your HW, fold your paper lengthwise and put your name on the top of
the outside.
Wednesday, April 2
Statements, Conjunction, Disjunction, Negation.
Read §2.1
Thursday, April 3
Read §2.2
Hypothesis, Conclusion, Inclusive or, the Conditional and the Biconditional
Friday, April 4
Tautology, Logically Equivalent, Logically Implies, Contradiction.
Self – Study: 2.1/7
Read §2.3
Comment: DO NOT use either of the symbols ⇔, ⇒, ≡ in a truth table.
Homework due Wednesday, April 9
2.2/ 4 a, b, c, 7 plus HW handout.
Monday, April 7
Go over exercise 2.1/7
Sets and Predicates
Read §2.3
Self – Study: 2.3/1
Tuesday, April 8
Truth Sets and the element form of a set.
Wednesday, April 9
Quantifiers
Thursday, April 10
Quiz & Group Worksheet (Worksheet is open book & open notes)
Quiz: 1.Truth Table 2. Tautology, Contradiction, Contingency
Worksheet:
1. Sequence of Equivalent Compound Forms (E.g., exercise 2.2/7)
2. Truth sets and element form sets. 3. True/False.