CS 198:352 Internet Technology
1st Midterm Summer 2006 (Time: 120 Minutes)
Name:
SID:
Section:
For TA use only: 1 2 3 4 5 6 7 8 Total
Do not open the exam until you are told to begin. Write your name on every page including your notes page. There is no
Natural deduction proofs
as in website (except False and
True are treated as part of the
logic rather than abbreviations
for p/\~p and ~(p/\~p) )
1
Borgida/Rosen 2016
SUMMARY: Natural Deduction Introduction
Rules that do not use assumptions
conjunction (
Disjunctive Normal Form
Disjunctive Normal Form
Theorem: every compound proposition S can be put in
disjunctive normal form.
Solution:
Construct the truth table for the proposition S.
For each row where S is true, conjoin an entry for every
proposition
REVIEW
Logical Equivalences Table 6 from book
ATA
AFA
Identity
Dominati A (BC) (A B) (A C)
on
A (BC) (A B) (A C)
AAA
AAA
Idempot
ent
(A B) A B
(A B) A B
De Morgans
( A) A
Double
negation
A (A B) A
A (A B) A
Absorption
A B BA
A B BA
Commut
ative
AAT
AAF
Sec$on Summary
! Predicates
! Variables
(K. Rosen notes for Ch. 1.4, 1.5
corrected and extended by A.Borgida)
! Quantiers
! Universal Quantier
! Existential Quantier
! Negating Quantiers
! De Morgan s Laws for Quantiers
! Translating English to Lo
ROSEN rules of equivalence for propositional and predicate logic:
! equivalence
$ equivalence
2-negation
DeMorgans Laws
Associativea
Commutativea
Distributive
Idempotence
Inverseb
Identity
Domination
Quantifier de Morgan
a Often
b Called
p!q p _ q
p$q (p!
Set Theory summary - Fall 2016
Prof. Alex Borgida
September 6, 2016
The textbooks chapter on Sets follows the one on Logic, and so relies on
its notation and results.
This document is a brief introduction to sets, which reflects my lectures
and which does
Propositional Logic 1 brief REVIEW
(section 1.1)
!Propositions
!Connectives
1.Negation ~
2.Conjunction /\
3.Disjunction \/
4.Implication
5.Biconditional
Precedence: Book says the order is 1 to 5 above. But the
only thing there is agreement about in books
Rules of inferences and Proofs with them!
(corrected and augmented slides!
from K. Rosen)!
Borgida/Rosen 2012!
1!
pTp
pFp
Identity
(p q) r p (q r)
(p q) r p (q r)
Associative
pTT
pFF
Dominati
on
p (q r) (p q) (p r)
p (q r) (p q) (p r)
Distributive
ppp
pp