Discrete Structures II CS 251
Lecture 2 (Sections 6.2.4, 6.3)
Review example: Reasoning (MP, MT, NS)
What rule of inference is used in each of the following arguments? 1. If it snows, then the roads are closed; the roads are not closed. Therefore, it doe
Discrete Structures II CS 251
Lecture 17 (Section 10.5)
1
Congruence
A congruence relation is an equivalence relation on the carrier of an algebra such that the operations of the algebra are preserved by the relation.
We can define an equivalence relation
Discrete Structures II CS 251
Lecture 16 (Section 10.4)
1
Relational Algebras
An algebra is called relational algebra if its carrier is a set of relations. Operations on relations are: select, project and join. relation; select, project, join
2
Operations
Discrete Structures II CS 251
Lecture 14 (Revise section 9.2, Section 10.2)
1
Revision: Resolution Rule (R)
Given the following two clauses. L1 Lk C and M1 Mn D, where Li and Mi are atoms and C and D are disjunctions of other literals. Assume also that 1.
Discrete Structures II CS 251
Lecture 13 (Sections 9.2, 10.1)
1
Section 9.2 Logic Programming
A logic program is a set of clauses with the restriction that there is exactly one positive literal in each clause. Such clauses are often called definite clause
Discrete Structures II CS 251
Lecture 10 (Revision)
1
Discuss chapters 6,7,8 of Hein, J. L., Student Study Guide for Discrete Structures, Logic, and Computability. Jones and Bartlett, 2003. Material will be handed out in class.
2
Discrete Structures II CS 251
Lecture 7 ( Examples, Section 8.1)
Chapter 8: Applied Logic
Can we formalize things that we talk about? For example 1. When we reason we geometry we make assumptions about points and lines. 2. When we reason about automobile
Discrete Structures II CS 251
Lecture 5 ( Examples, Section 7.2)
Example 1
Find examples of wffs with the given properties. 1. The variable x has three bound occurrences and one free occurrence. x p(x, x) q(x) 2. The variable x has four bound occurrences
Discrete Structures II CS 251
Lecture 4 ( Section 7.1)
Chapter 7
Predicate Logic: used to analyze a wider variety of arguments and statements than propositional logic. Example:
All computer science majors own a personal computer Socrates does not own a p
Discrete Structures II CS 251
Lecture 3 ( Examples, Sections 6.4, 7.1.1)
Review example: CP rule
Use CP rule to prove that the wff is a tautology. ( A B) (B C) ( C D) (A D). 1. ( A B) P 2. (B C) P 3. ( C D) P 4. A P start subproof 5. B 1, 4, DS 6. C 2, 5,
CS251 HW#6
1. Pg 530, Question 1(c,e,f). Use Skolems algorithm, to transform each wff into a clausal form.
1c. x y(p(x,y)q(x).
[Refer back of the book]
1e.xy(p(x,y) z q(x,y,z)
[Refer back of the book]
1f.x y z [( p(x,y) q(x,z) ) r(x,y,z)].
Ans:
Step1: Dis