Status: Tags: Links: notes/) MACM 101 - Discrete Mathematics I
MACM101 Slides
Yet to do
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Slides
2
Propositional Logic connectives
Practice
- No. 1 ,3, 4, 8a, 8c (page 54)
3
- conditional versions
- Logical equivalence
- Tautologies and contradictions
Practice
- No. 9, 13, 17 (*) (page 54) No. 1a i,iii (page 66)
4
Practice
- No. 1ai, 2, 6a, 6b, 14a (page 66) - Express conjunction and disjunction through implication and negation (*)
5
Practice
- No. 1a, 3c, 4a, 5c, 9a (page 84)
6
Conjunctive Normal Form (CNF) - Two rooms puzzle
Practice
- No. 5, 9a (page 84-85) - Prove that resolution is a valid rule of inference - Same arrangements as in the Two rooms puzzle This time if a lady is in Room I, then the sign on it is true, but if a tiger is in it, then the sign is false. If a lady is in Room II, then the sign on it is false, and if a tiger is in it, then the sign is true. Signs are
- I both rooms contain ladies
- II both rooms contain ladies
7
Open Statements or Predicates Universes Quantifiers
Practice
No. 1, 2, 4acij, 9a(i,iv), 12(vii,viii) (page 100-102 )
8
Open/Bound, Compounds Definitions/Rules/Theorems
Practice
No. 17ab, 18ac, 25a (page 100-102) 7b, 8b (page 116) Represent in symbolic form a definition “Jaywalk means to cross a roadway, not being a lane, at any place which is not within a crosswalk and which is less one block from an intersection at which traffic control signals are in operation”. a rule “No driver of a vehicle shall drive such vehicle on, over, or across any fire hose laid on any street or private road, unless directed so to do by the person in charge of such hose or a police officer”
9
10
Discrete Math Theorems Proving Theorems
- Rule of univ spec, univ gen
Practice
- No. 5, 9, 11, 13, 15, 17 (page 116-117)
11
12
Practice
- No. 1, 2, 4, 6, 8 (page 134)
- Give a formal proof of the theorem on slide 12-9.
13
Practice
- No. 1, 4, 6, 8bc, 16, 17bc (page 146 – 147)
14
Practice
- No. 1, 4, 5a (page 252) - Prove part (3) of the theorem on slide 13-14
15
Equivalence Relations - Congruences - Partial Orders
Practice
- Are the following relations reflexive? symmetric? transitive? antisymmetric? - Motherhood:
x is the mother of y’ - Intersect:straight lines x and y intersect’ - Show that the relation ⊆ on the power set of a set is an order. Draw the diagram of this relation on the power set P( { a, b, c } ).
- Which of the properties: reflexivity, symmetricity, transitivity, and anti-symmetricity, should be true for a relation expressing the idea of similarity (not necessarily identity)?
16
Practice
- No. 1, 2, 6a,15, 16ace, 18 (page 258)
- 278, 140
17
Practice
- Exercises from the Book: No. 1def, 2b, 4 (page A-32)
- Construct a bijective mapping between the closed interval [0;1] and the square [0;1] × [0;1]
18
Practice
- No. 1a, 2b (page 208)
- No 3, 4a, 7b, page 244
19
Practice
- No. 1b,2a, 14 (page 208-209)
20
Practice
- No. 1, 4, 11, 14, 24 (page 12)
21
Practice
- No. 1, 4, 11, 14, 24 (page 12)
- No. 2, 9, 12, 15 (page 24)
22
The Binomial Theorem - Pascal’s Triangle - Pascal’s Identity
Practice
- No. 22a, 29, 30 (page 25)
- No. 5a, 7, 10 (page 34)
23
Practice
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Exercises from the Book: No. 4, 10 (see Example 5.45, p.275), 14, 18 (page 277)
- Determine Ramsey number R(4,4) (difficult)
24
Practice
- Exercises from the Book: No. 1, 5, 9, 15 (page 156)
- 1, 4, 7 (page 164 – 165)
25
Integers - Division - Division Algorithm - Prime Numbers - Composite Numbers - Binary - Hexadecimal
Practice
- No. 2, 4, 12, 15, 16 (page 230)
26
Greatest Common Divisor - Euclidean Algorithm
Practice
- 1ab,4,5,10,15 (237)
- 1ab,5,7,9 (241), 123
27
Congruences - Residues - Caesar Cipher - Psuedorandom Generators
Practice
- No. 1, 5, 9, 12, 20, 23 (page 696)
28
Chinese Remainder Theorem - Linear Congruences
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Created:: 2021-10-05 12:32