NED University of Engineering & Technology, Karachi
algebra
MATH 12394

Spring 2017
Flying Tiger Written Report
Michael Champion
Meghna Shambharkar
Henry T. James
Ben Okpala
Western Governors University
FLYING TIGER WRITTEN REPORT
D1. Financial Statistics
Analysis of key financial indicators  ROS, ROA, ROE, Leverage, Sales, and Profits:
NED University of Engineering & Technology, Karachi
Intermediate programming
MATH 101

Spring 2017
CIS 324 SP 17 Course Schedule
Wk
1
Week of
9 Jan
Content
Introduction & Syllabus, Ch 1
Topics: Intro to Database, Create Accounts in
SQL Exercise System
Assignments Due (check calendar for dates)
Read Ch 1
Create Account in SQL Exercise System
Start SQ
NED University of Engineering & Technology, Karachi
Intermediate programming
MATH 101

Spring 2017
University of South Alabama
Department of Communication
Term: Spring 2017
Course: CA 275101 DecisionMaking Small Groups
Class Meeting Time: Tuesdays and Thursdays 1:252:15 p.m.
Class Location: Communication Building, Conference Room,
USA Drive West
Ins
NED University of Engineering & Technology, Karachi
calculus 2
MATH 1342

Fall 2013
F RAMES I: L OGICS
WITH
D ISTRIBUTION
CONS
1999/11/6
page 245
245
E XAMPLE 11.12 (L ANGUAGE F RAMES WITH I DENTITY )
A language frame has the set of strings on some alphabet as its point set. A
language frame with identity includes the empty string . The
NED University of Engineering & Technology, Karachi
calculus 2
MATH 1342

Fall 2013
U SING S UBSTRUCTURAL L OGIC
CONS
1999/11/6
page 347
347
the conclusion false: in every world in which the premises are true, so is the
conclusion. However, in another sense, disjunctive syllogism is invalid: in some
states of affairs the premises are tr
NED University of Engineering & Technology, Karachi
calculus 2
MATH 1342

Fall 2013
M ANIPULATING P ROPOSITIONAL S TRUCTURES
CONS
1999/11/6
page 197
197
T HEOREM 9.26 (W HEN Ideal(P) IS C OMPLETELY D ISTRIBUTIVE )
If P contains no counterexample to (finite) distribution, then Ideal(P) is completely distributive.
That is, if P is a distr
NED University of Engineering & Technology, Karachi
calculus 2
MATH 1342

Fall 2013
242
CONS
1999/11/6
page 242
F RAMES
T is a right truth set for R if and only if for each x, y P, x y if and only
if for some z T , Rxzy.
The motivation of this definition is simple: if t is a left identity for fusion, then
it will be true at a left trut
NED University of Engineering & Technology, Karachi
calculus 2
MATH 1342

Fall 2013
230
CONS
1999/11/6
page 230
P ROPOSITIONAL S TRUCTURES
Conversely, if F is any set with these two properties, then define FF by setting
FF (a) = cfw_y B : a a where (a , y) F
So we can represent continuous functions by their traces. In fact, if F is co
NED University of Engineering & Technology, Karachi
calculus 2
MATH 1342

Fall 2013
F RAMES I: L OGICS
b
WITH
D ISTRIBUTION
239
a
c
a
CONS
1999/11/6
page 239
b
c
Figure 11.1: A Compatibility Frame
and x x, then there can be no clash between x and y, as x is included
in x. The same holds if y y. Figure 11.1 is a particular example of thi
NED University of Engineering & Technology, Karachi
calculus 2
MATH 1342

Fall 2013
200
CONS
1999/11/6
page 200
P ROPOSITIONAL S TRUCTURES
!
"
Conversely, suppose
that z i Ii . We wish to show that z i Ii . If
!
y Ii then I i Ii , so z ! y, so z Ii for each i, as desired.
The cases for the other connectives are similar and are left as e
NED University of Engineering & Technology, Karachi
calculus 2
MATH 1342

Fall 2013
F RAMES I: L OGICS
WITH
D ISTRIBUTION
CONS
1999/11/6
page 263
263
Now for (I; E), we wish to show that if Ryzw and xCw then there is a
prime theory v where Rywv and zCv. To do this, we consider whether zC(y; w).
Suppose that A yw. Is A z? Well, since A y
NED University of Engineering & Technology, Karachi
calculus 2
MATH 1342

Fall 2013
M ANIPULATING P ROPOSITIONAL S TRUCTURES
CONS
1999/11/6
page 203
203
Similarly, a casebycase analysis shows that x y ! z if and only if x !
y z. First, consider the case where x, y Q. Then x y = and x y ! z.
Then y z = when z P and we have x ! y z. Whe
NED University of Engineering & Technology, Karachi
calculus 2
MATH 1342

Fall 2013
CATEGORIES
CONS
1999/11/6
page 221
221
Just as with products, any two sums of an object are isomorphic. (The proof of
this fact is left to Exercise 10.3.)
D EFINITION 10.20 (C ARTESIAN CATEGORIES )
A category is cartesian if every pair of objects has bot
NED University of Engineering & Technology, Karachi
calculus 2
MATH 1342

Fall 2013
D EFINING P ROPOSITIONAL S TRUCTURES
CONS
1999/11/6
page 185
185
By induction, we can verify that the value [A] is one of b and b, and similarly, the value [B] is one of c or c. Therefore, [A] ! [B], and since this is a
model of R, we have A B in R, and
NED University of Engineering & Technology, Karachi
calculus 2
MATH 1342

Fall 2013
CONS
1999/11/6
page 323
Chapter 15
Undecidability
Things so utterly undetermined,
that they are indeed altogether undecidable.
Bishop Joseph Hall, Episcopacie by Divine Right Asserted, 1640
In this chapter, we will examine some of the most surprising re
NED University of Engineering & Technology, Karachi
calculus 2
MATH 1342

Fall 2013
194
CONS
1999/11/6
page 194
P ROPOSITIONAL S TRUCTURES
It turns out, then, that the class of propositional structures fit for a logic is
(almost always) a variety.
D EFINITION 9.17 (F REE S TRUCTURES )
Given a quasivariety V of propositional structures,
NED University of Engineering & Technology, Karachi
calculus 2
MATH 1342

Fall 2013
320
CONS
1999/11/6
page 320
D ECIDABILITY
P ROOF The verification is an induction on the construction of A. It holds for
atoms by definition. The induction steps for conjunction and disjunction are
immediate. Consider ! formulae. If x " !A then x " !A if
NED University of Engineering & Technology, Karachi
calculus 2
MATH 1342

Fall 2013
CONS
1999/11/6
page 308
308
F RAMES
P ROOF Soundess is trivial, as the ternary accessibility relation on language
frames satisfies B and Bc . Completeness is less trivial. We cannot use the canonical model, as the canonical model is nothing like a langu
NED University of Engineering & Technology, Karachi
calculus 2
MATH 1342

Fall 2013
F RAMES II: L OGICS R EJECTING D ISTRIBUTION
CONS
1999/11/6
page 287
287
R, S and C, so defined, are accessibility relations.
The conditions corresponding to the structural rules of S hold in the frame.
For each p, [p] = cfw_A : A p is closed.
For each A
NED University of Engineering & Technology, Karachi
calculus 2
MATH 1342

Fall 2013
206
CONS
1999/11/6
page 206
P ROPOSITIONAL S TRUCTURES
P ROOF This is a straightforward verification of the conditions for an S4 necessity. We will consider a few and leave the rest to the reader. (We will write !
for !O .)
!
!
!x !y = cfw_z : z " x and
NED University of Engineering & Technology, Karachi
calculus 2
MATH 1342

Fall 2013
CATEGORIES
CONS
1999/11/6
page 227
227
which is natural in A, B and C (draw the appropriate diagram yourself) and
satisfies the following pentagonal condition.
A (B (C D)
(A B) (C D)
A,B,CD
(A B) C) D
AB,C,D
idA B,C,D
A,B,C idD
A (B C) D)
(A (B C) D
A
NED University of Engineering & Technology, Karachi
calculus 2
MATH 1342

Fall 2013
F RAMES II: L OGICS R EJECTING D ISTRIBUTION
CONS
1999/11/6
page 281
281
D EFINITION 12.7 (M ODELLING L ATTICE C ONNECTIVES )
A relation ! between points in P and formulae respects lattice connectives if and
only if the following conditions are satisfied
NED University of Engineering & Technology, Karachi
calculus 2
MATH 1342

Fall 2013
CONS
1999/11/6
page 278
278
F RAMES
!
So, suppose that y(xCy z(yCz
z i i ). It follows
! that x i for
!
each i, and therefore that x i i as desired. Therefore i i is Cclosed, as
we had wished.
As with Beth frames, a disjunction is true not only at the
NED University of Engineering & Technology, Karachi
calculus 2
MATH 1342

Fall 2013
224
CONS
1999/11/6
page 224
P ROPOSITIONAL S TRUCTURES
10.3 Intensional Connectives
To model the other connectives of our logics in categories, we need to examine
one new categorytheoretic concept: the concept of adjunction. Let us start with
an example
NED University of Engineering & Technology, Karachi
calculus 2
MATH 1342

Fall 2013
236
CONS
1999/11/6
page 236
F RAMES
11.1 Examples
Before launching into a formal definition of a frame for a substructural logic, we
will consider a number of examples to motivate the definitions of the next section. The examples hark back to the motivat
NED University of Engineering & Technology, Karachi
calculus 2
MATH 1342

Fall 2013
CONS
1999/11/6
page 293
F RAMES II: L OGICS R EJECTING D ISTRIBUTION
293
of where a proposition is true and another to keep track of where it is false [4].
This is also a model of a substructural logics, but it is out of our ken in this
chapter.
Bell gi
NED University of Engineering & Technology, Karachi
calculus 2
MATH 1342

Fall 2013
170
CONS
1999/11/6
page 170
P ROPOSITIONAL S TRUCTURES
D EFINITION 8.36 (S TRUCTURES F IT FOR A L OGIC )
Given the system S, the structure P is fit for S if it is fit for the language of S,
and in addition, the translations of the rules of S are satisfie
NED University of Engineering & Technology, Karachi
calculus 2
MATH 1342

Fall 2013
218
CONS
1999/11/6
page 218
P ROPOSITIONAL S TRUCTURES
M this class is equal to (yC , M ), as we desired. The compofst M, N
sition of pairing of arrows with left projection gives the left component of the
pair. Similarly, the composition with right pro