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TEXAS A&M UNIVERSITY-CORPUS CHRISTI / Math / Mat 2305 / What is the difference between proposition and predicate?

What is the difference between proposition and predicate?

What is the difference between proposition and predicate?

Description

School: Texas A&M University-Corpus Christi
Department: Math
Course: Discrete Mathematics I
Professor: David thomas
Term: Fall 2016
Tags: Discrete, Math, Discrete math, logic, and formal logic
Cost: 50
Name: MATH 2305 Exam 1 Review
Description: Here is the study guide! All of the practice problems are solved, including his method for solving in class if it was different from my method. Quick reviews of topics listed in the review packet are included. **** I expanded my proof explanation/justification for #5, so it should be very clear now!
Uploaded: 09/30/2016
14 Pages 35 Views 3 Unlocks
Reviews


MATH 2305


What is the difference between proposition and predicate?



Exam exam

renew

Review

practice problems

5) prove the argument valid

6 (apvqr

©

(~t)

e lapan) → (25)

o decide whether or not there

a number whoresquare equals 4 times the number itself. Explain: 3 xed [x2= 4x]


Explain why the sentence that you are currently reading is false. is not a proposition.



Don't forget about the age old question of Who is the king of crete?

• let X-4 (4) 2 = 4 (4)

16 = 16

nypotheses are assumed true

0~put 2 ~p

de Morgan's M.T. O and © M.P. © and © M.P. 0,0 and cut and


What is universal instantiation?



ova

(2) Explain why the sentence

that you are currently reading is false." is not a proposition. We also discuss several other topics like What are some of the reasons we first use assessment tools when first meeting a patient?

proposition: a sentence with

a truth valne a the sentence doesn't have Don't forget about the age old question of What is the chief goal of a journalist?

a truth value (ie, if the sentence is true, it's falser, Don't forget about the age old question of Why cellphones must be put away at all times in lab?

and vice versa)

(6) what is the difference between

"Propositions and predicate"?

• Propositions are just sentences

with otvИТИ yajи

• predicates have a variable,

become propositions when variables are replaced with If you want to learn more check out What form of carbon is radioactive?

avaine from the domain

(3) Statt de Morgan's laws:

(AB) = (CA) V (B)

• (A/B) = (~A) A (B) 14) Decide if the argument

(7) negate the logical expression: WAXED, 3 y&D, [P(X,Y) * Q(x)])

AXED, YyED (v [~P(x,y) VQ(x)]) EX ED, VY ED Ev (v p(x,y)^(~Q] Don't forget about the age old question of Who do we cook meat?

2 XED, YyEp [P(x,y) ^ ^Q(x)] (6) what is universal instantiation?

every sub. Of an element ED WIU be true in a universal argument

pf: VxED[PC] (M. I.?

Hopea Pq P*99 p/ Pvq

.

copval

(9) compare prop. and Pred. logic (11) thm: if kis odd MATH 2305

wing the concepts of T-tables, and mis even, Exam / proof methods, tautologies,

Review predicater, and propositions:

pfi 0.1e7 R-anodd number → Prop. logici

2 = 2at 1

• Proposentence, letters, TIF

let m = an even number

• can show proof through

m=2b tables (if there aren't too

1 k 2 + m 7 = (29+12+(26) 2 many letters), annotated

=4492749+1) +(463) listings or proof sequences

= 2 (202-29) + 1] + 2 (262) valid argumente: M.A.M.T. cut, generalization,

= (2n+1+2v Specialization, 1 transitivity

- 22+2v+1 = 2(n+v)41 tantology shown in t-ables → pred logici

pred: sentence, variable is

generic H2+ņa ir odd incinded, becomes a prop

©:, Theorem istrue when variables are

replaced by values (12) prove or disprove: t-tables won't work for

toimi if n is odd, Disodd Universal statements

• V=all values

pf: if n=5, nirodd 3- at least one valne

but 15-17 - 4 = 2 even

• truth Of Y can be shown

thronan U.I. and M. p.

•] only needs one true (13) repeating decimal 1.2323 val nie; can be found

as a rational number thronan guessing, testing all options when finite)

112 32 32 line up

I repeating vare tanto logies

100 X = 12312323) decimal theorem:

9 9 X = 1221 (10) prove that the square of an

odd number 1. odd:

X=122 >pfi

Olet =anodd number ~ 9 = 29+1 def. Of odd)

X = (20 + 1)2 = 492 +4+1 = 2(29+20)+1

(292+29) = b = 26+1 3 generic ý zis odd! es theorem is true

MATH230S exam Review

(14) thm: sum of any 3

I consecutive integers is divisible by 3

(17) Matching

terms may be used more than once, or not at all

Pf:

let â = an integer

ât câ+1)+(6+2) = 3 + 3 = 3(@+D (= 36 ) :

36 generic â makes theorem

TERMS:

B. variable C a proposition E cutlelimination FAB

de Morgan's laws ( modus ponens

NO Term appears

hence, theorem is true

(15) thm sum of any two rational

number is a rational

DEFINITIONS:

1. a formula which is logically

eanivalent to AVB

E. AB

of 9 let

a

and C be rational

2. ~(AMB) = ~AVNB and

~ (AVB) = ~ANNB are called

I. de Morgan's laws

c. ad bc

=m

3. a name used for an unknown val

variable

bd to because bad to

4. place where variable values

may be found

NO Term Appears ► domain

theorem works for generic I values (a, b, c, d ) ☺ hence, theorem is true

5. meinod to conclude "Bmust be

true when "A" and "AB" are true

modusponens

(16) what is wrong with this "Proof"

ihm for any integer ko

K2+2K+L is a composite [Pf 22 +222 +1=9 and 9 = 3*31

6. Valid proof that can be used to

implement M.P. and M.1. © Cut/elimination

thm. says "any integer, meaning it must be

universally quontified (1), not just one example. They should have ured a

generic value for ķ

7. A → A is this type of formula

statement form

No term appears - implication

pencil = proof

green commentary red: justifications

(5) prove that the following argument is valid hypotheses (@lup v

q r are labeled o sv (9)

. this is This is

our @-@,50 L can easily

theorem refer back in proof

Le (Par) (us)

. (9)

1 @ pt

Proofi

Justification' @ these are our

hypotheses from above.) all hypotheses

• I'm just too lazy to } are assumed e rewrite them *

true o uput

Togically equivalent to a (Not demorgan's. I wrote

@ the wrong thing before )

@pt = ~puto 2 ~P

modus tollens using 0 and ©

orovt

@rt 3 :. ~P

wpvq)

I talked to Dr. Thomas and he

said my Original Mełnod needed the explicit addition of these steps

generalization using ☺

3 up ®:.,(npVq) Modus Ponens using 3 and 4

@ (~PV q ) ►r

3 (~Pvq) Or specialization using @ and ☺

I

tassumed Lovern

them before)

VPNO

2 up

VOUSOU

©ns

0 :o) (wpar Modus ponens using @ and ☺

@ (upar) ~ 6 (~Par)

anos pris

Study Sour

SOU

SEU

SOU

Hope this helps

bro

sro (л) AS ®

the theorem! conclusion from

ЛИО ЈэлэИ

@ Pub @ buisn UOLTOU! Wilə) tha

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OS APS

math 2305 10.4.16

alternative solutions (9) compare prop and pred

logici (Prop. pred.

usually 10-20 matching

(definitions)

Truin tables

prop. logic only not in pred. logic

- X2 4X=0

X (X-4)=0 X=0 X-4=0

X=0 or X = 4

stable

(2) can't establish a

truth value

Tantologie

always true prop.: checked through

truth tables for all Possible truth value assignments, the

formula is true

•Pred for all possible.

pred. definitions n all possible domains

FX ED [P(X) V ~P(x)]

(3) ~YXED[ ] = 7XED ~[ ]

waXE DE 1= AXED [ ] (can include these universal

oner if you want)

(4)

-

T p q lettersdon't need to be T q pf D true for the line flio, pval to be true

predicates;

ca senience, contains

variables, becomes a es proposition when variables are replaced by values P-name (x, V, Z)

A() = a prop. in pred. logic proposition

a sentence that has exactly one truth Value A inse a letter to signiful

(51 0 up

Mite © and 2 ~pva) generalization and

M.P. @ and a 9 (up 1r) specialization 0 and

us

Mip. and

cut and b - work: opet ☺ (prqder 3 (upvq) ar

vt vp

fupva) up (vPvgi ir --P

p^n + SV69) heart

1 ~q

10.4.16

alternative solutions conta... (1) Olet ke an odd number (14) ► problems like this won't be

land me an even number on the exam. 2 k = 2atl. atz I m = 26. bez

(15) Pf: let rur EQ crational)

r. = 9/6 anbez, bto = 492 +49 +1 +4b2

- P ă cadez, d to 2.[222 +29 +262)+1

3 a C - [ac] - m mnez = C, CEZ

b d [bo] n nto t = 2C+1 Caef. Of oad)

Ifine producte 2 generic k24 m2 isoad

numbers is o, one hence inlorem is true

or both must be o

*on previous review solved for (12) prove or disprove

sum of 2 rationais, problem

15 says Product! if trueshow

- Itaire,

OOPS Proof

give a

Counterexample (16) cant substitute ("proof by example") expression (thm if n is oad, then

for infinitely many possible - values

universal

Ipf:

Counter example

(17) 7. A A is a tantology

_ ®NO Term appears

n = 5

151 - 4,2 leven)

2

21

ithm is false

"the first time it's La thick, the rest

. Of the time it's at ChiИИС"

MATH 2305 exami Review

set-builder notation

S = x.suon that Pex is true}

S = EXP(x) abbreviated

CHAPTER 1: Speaking mathematically

1. variables

· used for unknown values ex: is there a number such

that doubling it and adding 3 gives the same

result as sanaring it?

2x+ 3 = x2

X-3 or X=-1 used for arbitrary generic values ("all" values) ex' if a number is >2,

its square is >4 > 107 X= "a numbet."

if X2, X274 #used in proofs

• universal statements: "for

all/every x ino"

• conditional Statements:

if O, then 4"

AS B if yea, then XeB YEB, but YEA Russein's paradox: problem

wim Sct-builder notation

{XIXX3=B if BEB→ B&B

BOB >> BEB → repair by assigning a

universe (U) B = { x} X3 B + 6 because

640 (B must not

be in the universe) cartesian Products

ordered pairs

(a, b) = (0, d) iff a: c^=d.

AXB-ga,baeA, beB3 1.3: language of reiationen functions

. dof: reation between sets

used to describe non-functions ex: a circle

*9) y-values are On not unique

1.2 Language of sets

Notations E "in"/"isa member of "

ex: XED "not in"/"not a member of

ex: x € 0 subset ex: A={1,2,33 B={1,2,3,4}

ASB

• useful sets

Z inteqers -1,0,1,2...3

relation

z+ Positive integers {1,2,3...3 set roster notation

S={1,2,33

Lohisiory of members Zt={1,2,3... B infinitely

" def: function

given Xfex is unique every XED must be used

CHAPTER 2: 1091c Of Compound

statements 2.1: logical form and logical

equivalence

def: proposition

a sentence

lear truth value (TED wysymbolized by a letter

logical operators ^ "and" / "conjunction" V orld is junction" ~ "not" negation truth tables shows fruin

values of prons lex: ALB AVB AAB

MATH 2305 exami

Review . de Morgan's laws

MAVB(MAA(UB)

N (AMB) = A (B) 2.2.conditional statements

deft implicanon ()

- if A is true, then B is true - if A is false, the

implication is true I by defain it

ABA-B

10gical equivalences * AB & B-A

NOT THE SAME! * ABS (VA)VB

AB AB (VA)VB.B TA

TE FT * Syntax and semantics

binary needs two Props (ANB). binary unilateral (one prop

(A) tautology: for all possible truth asian ments, the argument is true ex AMA AVA)

contradicion for all possible

truth assignments, the arqument il faise lex A LA CAMA

negating conditional

statements N (A B ) og. equiv

M (AVBV becomes 1 ( VA)^(^B) ariveinn

An (B) double

negation

• converse: interchange

hup and conc.

ex: A B becomes

logical equivalences BAA) (AAB) (AUB) = (BVA)

"If it rains, then they

cancel school"

becomes Mif they cancel senool,

then it rains

--> prove by Cheding the

truin table

- argument fom cont'dono

A B nypothues

A

MATH 230S exam! Review

inverse: negate H and C 16PQ) becomes

ploQ)

if it rains, then they cancer school"

becomes "If it doesn't rain, then

they don't cancel school" Contra positive interchange

the inverse (P-Q) becomes

B6 B conclusion

def: vaid arguments when the hup. are all true, the conclusion is always true testing for validity

A make a t-table and

I check it annotated listing of

argument © ure a proof sequence * VALID ARGUMENTS:

MODUS ponens

→ Modus Tollens

Mifit rains, then they

cancel school

becomes Wif they do not cancer school,

then it didn't rain"

• def: biconditional (*)

boin props must have the same truth valve

(if and only if /"iff")

A3 A**B

-VB

-> cuti elimination

AVB NA

→ Generalization

O, A NE → Specialization

• Order of performing operations

negation 2) AV coni./disi. 3) imp. 'bi cond.

[GA) - (Bņ(yg)]

franshivity

A TMB

--> proof bul cases

IPVQ

2.3. Valid and invalid arguments

def: Grayment forma

a list of 109109l expressions in symbolic 10916

. Proof sequence example:

sep

TA

hyp. are assumed

true Mat. using @ and a

oup

@~5

MT. using and

MATH2305

exam,

Review

• def: universallu quantified

statement (A) VXED [P(x)] "Ifor all x valves in D,

P(x) i frue" def: existentially quantified

Statement (3) IXED [PCD] for at least one x ind

P(x) is true" formal v. informal

language → informal human lang,

formal: symbolic lang, FORMAL → INFORMAL

is easy INFORMAL > FORMAL

is hard

• implicit quantification "integers, are rational #s"

La universally quantified

even though we didn't explicitly say

all integersen 3.2. Negating Quantitied

Startements Theorem 3.2.1:

to (VXED [Q(x)]

EXED ~ Q(x37 Theorem 3.2.2 MEXDCQ(x)])

or

cut using

Irus

and

Crot

M.p. using

and ©

* weaknesses of prop. logic

→ no universala statements;

the closest is listing

all Objects (n) →no "existential" statements

without finding an

explicit object

CHAPTER 3: Logid Of Quantified

statements 3.1: Intro. to quantifiers and pred.

• def: predicate

sentence becomes a prop. When variables arereplaced by valueSED

Negating universal

conditional statements

(Vxeb [P(x) = Q(x)]) = 3XED ~ [~P(XVÕx2] LEXED LP(x) V ( QO))]

~ (EXED [P(x) + Q(x)]) = VXED ~ EvP(X) VQ (x)] = WED [POV (Q(x)]

negations of multiply-quantified statements ~ VXED CEYED [P(x) 12 E ]XED IYED [PX) = 3 XEDLEYED INP (x, y)]] Order Of Operators → doesn't matter when

all quantifiers are & ex: VXED VY ED [P(x,y??

= VYEDVXED[P(x,y)]

MATH2305 exami Review

• relationship to de Morgan's when domain is finite v is a generalization

of and statements ] is a generalization

of or statements

• variants Of conditional

Statements (ex : 2.2.5) ~ "if sarg lives in Athens

then she lives in Greece" = 'sara lives in Athens, and

she does not live in Greece she could live in Athens,

Ohio or Athens, GA) 3.3: Statements with multiple

quantifiers ex: "Somebody (3) Supervises

every (v) detail"

3XED VYED [sup.(x, y)] person détail people: * *

3.4: Arguments with Quantified

statements o def: universal instantiation

(YX EDLA(x) - BOX] MA(8), XED

68 B (82

→any substitute of an element in Dis true

by definition нх ухер Грих) C & PC2, XED

proving validity of arguments wiquantified Statements

betails * * * * * *VY Hevery detail has somebody sup."

VYED [IXED[Sup(x, y) detail person people: * * * *

[details:

* ORDER IS IMPORTANT! informal to formal logic "There is a smallest

Positive integer," → one quantifier ]

2X€ 7+ smallest XI -> multiple quantifiers

ZX67+WVE 7+ (x=y) it's hard to move from ambiguous informai io concrete forma

E

negations of multiply-quantified statements ~(VXEP (ayed [P(x)]).

XED []ye D[PX EXED VYED (up (y)]] order of operators" → doesn't matter when

all quantifiers are t ex: EXED VYED [P(x, y]

= VyEDVXED[Play]

dy Soup

Studysoup

tour

MATH 7305 exami Review

• relationship to de Morgan's when domain is finite

V is a generalization I of and statements.

I is a generalization

of or statements

• variants of conditional

Statements (ex : 2.2.5) ~"ifsara lives in Athens,

then she lives in Greece" = sara lives in Athens, and

She does not live in Greece Che could live in Athens,

Ohio on Athens, GA) 3.3: Statements with multiple

quantifiers

• ex: "Somebody (7) Supervises

every (v) detail".

]XED[VVED [sup.(x, y)]] person détail people: * *

13.1: Arguments with Quantified

I statements

•def universal instantiation

wVX ED LACX) - BCX)] HEA(R), RED

→ any substitute of an

element in Dis true | by definition H?VXED [P(x) C{ P(8) XED

proving validity of arguments wiquantified statements: use universal instantiation to solve for generic valves → universal modus ponens (UMP)

universal modus tollens (UMIZ

TXED [PCX) + Qx)] ware). RED

bletails * & ** **vy Fevery detail has somebody sup."

VyED [IXED Sup(x, y) detail person people * * * *

Isoup

details: *

* ORDER IS IMPORTANT! informal to formal logic "there is a smallest,

CHAPTER 4: Elementary Number Theory

and Methods of Proof

1.1 direct proofs and counterexample

def even integer : x 2a, atz

loddínteger x+26+1, bez

Prime in tegec: p2,

P-a*b, anbez, avb=! composite integeri CE Q*b, bue anbez, and to

positive integer." → one quantifier

Xe 7 challet (0) multiple quantifiers

IXEZ+VYE 7+ (X=Y) it's hard to move from ambiguou informal to concrete formal

Study So

) studySoup

Soup

getting proofs

MATH2305 Started:

exam = look for familiar Review

language in the

theorem lexithm every complete

bipartie graph, is

connected

Stugu

pf:

let G be a "complete bipartie graphia

a ou

. proving existential statements → find a value, by any method where it is

tre guessing.computing > or snow that is

impossible disproving universal Matements by counterexample: if we can find even one instance where thm. isnt

true, it is false proving universal statements I use an arbitrary value. and the definitions of lodd, eren, etc. to prove memod of direct proof

→]: find one suitable item → V, finite D method of

exhaustion (test every

Possible value in o) → V infinite D: show thm.

true with an arbitrary value (x) to prove suggestions for writing

proofs : o copy thm accurately

"thm: 3 mark beginning of proof

© Hence, Gis

connected 14.2: Pational Numbers

defi rational number

REP, PAQEZ, Q = 0

irrationall number

irt e, Page 7, qt0

theorem 4.2.1 sum of rationals is rational! ef: 9 letr. and he be rational

r = al anbet, bto

I r =

codez, dzo

☺ define all symbols carefully o use complete sentences 6 justify statements

© use therefore, hence, etc.

• common errors:

proof by example in an linfinite domain 2 using the symbol for

2 + ideas 3 conclusions with out

support 4 assuming what is to

be proved

3 ritra= 9.c - adtbc

be at bd = fE Z e enfet, fto

blcbndto © sum of generic numbers

is rational ☺ nence, theorem is true

• misc properties of rationais:

→ sum of rationals is

rational 7 product of rationals

is rational

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