Professor Kate Poirier, Spring 2017
a. Determine the prime factorization of 1078 and 2541
1078= 1078/2= 539/7 = 77/11 = 7
= 1078 = 11*7^2*2
2541 = 2541/3= 847/7 = 121/11 = 11
= 2541= 11^2*7*3
b) determne the greatest common divisor
the common divisors are 3, 7, 11, also 77 because 7*11= 77
77= gcd(1078,2541)
c) determine the least common multiple
the common denominators 3 , 7, 11
3 = lcm(1078,2541)
DL YLHSSF SPRL KPZJYLAL ZAYBABYLZ HUL HSNUVYPAOTZ. PAZ MBU AV ZWLHRPU JVKL.
a=superman is able to prevent evil
b= superman is impotent
c=superman is willing to prevent evil
d=superman is malevolent
e= superman exsists
f= superman prevents evil
argument form=
(a∧c)→f
¬a→b
¬c→d
¬f
e→¬b∧¬d
___________________
:.¬e
Argument =
(we have to prove ¬e)
¬a→b
¬c→d
e→¬(b∧d)
*i am lost after this
i know this is supposed to look like this
and pg 73 of the book has it but i dont really understand how the substituting works
this website explains more about building arguments:
http://sites.millersville.edu/bikenaga/math-proof/rules-of-inference/rules-of-inference.html
3a
T(a,b) = Student a likes TV show b where the domain for a is the students in the class and the domain for b is the shows currently airing on TV
Translate:
∀s∃t∃u(T(s,u)→¬T(t,u))
T(s,u) = all students s like a TV show u
or everybody likes a TV show
¬T(t,u) = it is not the case that a student t likes the TV show u
its not true that one student likes the TV show
(T(s,u)→¬T(t,u))
if everybody likes a TV show, then its not true that one student likes the TV show,
* can this be translated into :
if everybody likes a TV show, then there is not one student who doesnt like the TV show?
3b
use quantifiers and variable to translate the following prop
there is a TV show that everybody likes.
on the test i put this:
∀x∃y (T(x,y))
*should it be
∃y∀x (T(x,y))?
January 31 2017
ch 1 the fundamentals- logic and proofs
1.1 propositional logic
point give precise meaning to math statements
applications
-design of computer circuts see sec 1.2
-writing of software
-verification of correctness
-computer programs
-etc
proposition a statement that makes a declaration -that can be true or false but there is no burden to proove truth
examples
p= “2+2=4” T
q = “the sun comes up every day”
r = “the world is flat”
america is the best country – doesnt matter if its an opinion
2+2 = dog
non examples
any question
where are we
commands
make me a taco
2+2
building new proposions from odl propositions
1- negation (!P)
if p is a proposition
¬p means ‘not p’
or it is not the case
¬p = “it is not the case that 2+2=4” F
if p and q are propositions
p˄q means ‘p and q’
2+2=4 and the sun will come up every day
p q
| P | Q | p˄q |
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | F |
id p and q are 2 props
p˅q means ‘p or q’
2+2=4 or the sun comes up every day
in english can mean either a or b but not both (exclusive or) and a or b or both – inclusive or
| P | Q | p˄q | p˅q(inclusive or) |
| T | T | T | T |
| T | F | F | T |
| F | T | F | T |
| F | F | F | F |
p⊕q means ‘p or q’ but also not both
p⊕q ‘ 2+2=4 or the sun comes up every day
can be shown ˅
| P | Q | p˄q | p˅q(inclusive or) | p⊕q
(exclusive or) |
| T | T | T | T | |
| T | F | F | T | |
| F | T | F | T | |
| F | F | F | F |
p→q means ‘if p then q’
if p is true then q is true
if 2+2 = 4 then the sun comes up every day
p implies q
if p is true q must be true
if q is true it doesnt rely on p for the statement to be true
| P | Q | p˄q | p˅q(inclusive or) | p⊕q
(exclusive or) |
p→q |
| T | T | T | T | F | T |
| T | F | F | T | T | F |
| F | T | F | T | T | T |
| F | F | F | F | F | T |
p↔q
(p → q) ∧ (q → p)
p if and only if q
this means
“(if p is true then q is true) and (if p is true then q is true)”
| P | Q | p˄q | p˅q(inclusive or) | p⊕q
(exclusive or) |
p→q | p↔q |
| T | T | T | T | F | T | T |
| T | F | F | T | T | F | F |
| F | T | F | T | T | T | F |
| F | F | F | F | F | T | T |
THE ORDER OD 5 AND 6 MATTERS THE FIRST IS DEPENDANT ON THE LAST
7 Q→P converse of p→q
8 ¬q →¬p contrapositive of p→q
9 ¬p→¬q inverse of p→q
| p˄q | 7 | 8 | 9 |
| T | ? | t | ? |
| F | ? | f | ? |
Tell us anything you’d like us to know about you! Some suggestions are…
My major is computer systems technology, i was enrolled in advanced science classes since grade school, and always participated in science fairs, eventually going to intel in high school. i chose to work in theatre doing lighting though, but when i realized i disliked this career choice i re-enrolled in citytech and switched my major from theatre. i hated my last logic course but im hoping this will be better and i will understand the course. i do like calculus so maybe. my other hobbies are riding horses (i have worked at stables in brooklyn) and traveling when i can.
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