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UTA - MATH 1301 - Class Notes - Week 3

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Math 1301

Logic: Simple Statements

and Negations

Obiectives

lnterpret and express simple statements in

symbolic form.

Express negations of statements.

lnterpret and express compound

statements in symbolic form using

connectors.

Statement: has a truth value

Texas is the largest state. (p) F Oklahoma borders Texas. (q) 1

Negation: (-)

Texas is nr* the largest state.(^p )T 1,.

Oklahoma c\urr nr\ border Texas. (^ .1 )tr 2. If you want to learn more check out What is the difference between substantive law and procedural law?

Watch quantifiers, (all, none, some) especially when negating.

All fish can fly. (f)

(- f) ? Sovn( .t \ s\^ Lov"rnu\ t\v\ Some fish can fly. (s)

(-s) ? No tttvr Cu\n t\\4

No fish can fly. (n)

(-n) ? Sunre tc,t.vr t\$

Some fish cannot fly. (c)

(-c) ? n\\ $tttn Ltrvl t\9

All are .1. Some are Don't forget about the age old question of Where is olmec located?

None are

Some are not

Coniunction: (A) and, but, howeveq nevertheless

Texas is the largest state and Oklahoma bordersTexas.( Pnq )

Texas isn't the largest state but

Oklahoma borders Texas. ( - p^ q )

Disjunction: (v) or

Let s = lt is snowing.

Let r = The roads are icy. Don't forget about the age old question of What are the valve problems?

1,. lt is snowing and the roads are icy. S A R 2. lt is not snowing and the roads are not icysn 3. Either the roads are icy or it is snowing. rvs 4. lt is not true that, it is snowing and the.^.(SAr roads are icy.

5. lt isn't snowing but the roads are icy. n" SnY<We also discuss several other topics like What is paranormal technology?

Two final connectors:

If"..then {conditlqqal} -)

Antecedent )Consequent l{

-thtn_

If it snows today then the roads are icy. S -1'- "\F S th{nY. "

It snows today if and onlv if the roads a re icy. $ a-1 'r

Formal Name Symbol Read $ymbolic Form lltr{re! t*Not" *l) Conjunction n "And'n phq I Disjunction V "0r"' pvq ; Conditional ---+ "lt"-then" p "'+ tt I Biconditional <.-r "lf and only if" P#qWe also discuss several other topics like What is an another name for proteins?

Lets=Youstudy.

Let g = You get good grades.

Letc=Youearnacredit.

1,. lf you study then you get good grades. 2. You get good grades -)3if and only if you study. q<-) s

S

3. lt is not the case that, if you study you do not get good grades' n- c s - l ^- g )

4. lf you study or you you earn a credit.

get good grades, then tsvq) C Don't forget about the age old question of Who discovered the neutron?

Let:

d=dinner at Antonio's s= includes soup

a= includes salad

o= includes potatoes p= includes pasta

c= includes carrots e= includes peas

c[ -: CCSvcr) n Covp)Atcve ))

Summary

Can you write simple statements in symbolic form?

u' Can you express negations of statements? * Do you feel comfortable writing compound sentences, using the

connectors and, s1; conditional and biconditional?

Math 1301

Truth Values and Truth Tables for *, n, V

Obiectives

' Construct truth tables and determine truth values for negations, conjunctions and d isj u nctio ns

Truth Table for Negation

p q

Sally plays the piano. \ :\utr vn<.n\ =1 L Co:e ) Roberto will sing.

Truth Table for Conjunction

3 sta1.*mr:) q t cL\e ) € xc^."nPl e -"

T T F

F

S0r\\.1 wr\\ t)\x".4

T

P\unu On cA

ho \ocrf0 s^t t\\

F

S rvrt4 "

T

F

O CO(] 3r.J n Lt uvy.r " Cl v.0\ "

w\rtn hxrtvr Purrt- o\rc trut. Cn\q ttm-e \\ t rve \\ Truth Table for Disjunction

P = Sally plays the piano.

or

Q = Roberto will sing.

* $"tq t\vv1,.n

U d\l1uv'rLttt,o

T T F

F

T

F

T F

\\0t^'t \\ tu\\( \) W141fl hutvr 0K tL\\ \c

Determine truth values for p V *Q

T T F

F

T

F

T F

How many cases possible?

Ast nl.*rnn =) Q cq)&l

nZ

What if there were 3 statements to t&t? Aza6

TT-1 t\\- TT T TFT

€ T.T rT F f f T F rr

r

Fr

Fa T.I l

{-

FTFTfrFf TTr1TFtr

T

T -T

€

T T

T F;\

T

(

T

F

T F

T \-

Hours af Sleep

5 hours" 4 or ferver.

True or False?

It is false that 30% of American, g*.t O hours of sleep each night@%get5 hours of sleep dach night.

4-c

F ^a) F-rn (v

Summary

. Construct truth tables for compound statements made of 1, 2, dnd 3- statements?

' Find the truth value of a single case without a table?

Math 1301 ,l ,1

Truth Tables, If-then and Iff

Obiectives

* Construct truth tables and values of statements using and biconditionals.

determine truth cond itiona ls

Conditiona I (lf-then)

p = You buy dinner.

TT TF FT FF

-T

F

T

T

Q = | take you to the movies.

t+ \ou buS C\nne r thrn \ ttrrtt qAU \o rns{ gi

When is a conditional hrht vr C^

fa lse?

Jrw"\t"-lto\r("

TT TF FT FF

T F

T F

F

T T T

Tir :- -T,l

F]T

I

F i"i

r

T

F

T

x

Biconditiona l--lff

p = You buy

TT TF FT FF

.T

F

;.T

d in ner.

Q = | take you to the movies.

When is a biconditional true?

\4.) h e vr hotvr :Yo\Ye wLq-nrr

hmw tht \q\^^rr tY\!\\A VU\\)(

TT TF F, T FF

*i

rT

T

t

{

F T

I.T r-I

T T T T F

F

F

F

T T F

F

T T F

F

T

F

T

F

T

F

T F

Consider: If Spike Lee is a movie director or Halle Berry is a school teacher; then George Clooney is a circus clown.

d = Spike Lee is a movie director. t = Halle Berry is a school teacher. c = George Clooney is a circus clown.

T T T T F

F

F

F

T T F

F

T T F

F

T

F

T F

T

F

T F

Contradiction-answer column all F's Tautology-answer column all T's lmplication-all T's for a conditional

an A.A. Ilcgree T or F? lt is false that if 9

Credits Needed for

credits are needed for a

Cultural 'issues. 6

Communicafions degree or 4crediti are needed for a Humani-tjes degree, then

Humanities' 6 credits 5ie needed for a triathematics, Math degfee.

C cI'[)*:t TT

n- t -t-l

F

Math 1301

Equivalent Statements

Obiectives

. Determine the logical equivalence of statements.

p< )q isequivalentto

/o ) q)n (q ) p)

i

(p< >q) <=+[( p + q)A(q + p)l Same truth values in answer columns.

T T F

F

T

F

T F

T F

F

T

T T F

F

T F

T F

rT: I

F

r

ls ^,(p A q) equivalent to -p V -cN ?

T .:

T

,F F

T

F

T F

F

T T

,I

:rr r

F

F

{

T

T

F T T

tr -Tf T

What about the negation of a disjunction?

TT TF FT FF

F (

€

T

*r

-r

-T

tr

De Morgan's Laws

rf

T T

rrf T

F

Tf T

t. -(pAq)er*pV *q

Z. *(p v q)*+ *p A *q

filC coo\.C rs olut$= 1

1. My A/C does not cool and it isn't quiet. a cA*Q

{L ccvl)

Note that 2. The roast

- (- p) -p

h^d

is hot but it is not well done. hn *r)

L(*h) ^ ^cl

dtt Jrlnctwv^r *) 1- (, n'h V d )

TT T TF-T F. T f,- FF?

TTF

TFT

F.T { FFT

T

tr

Tl

T-r

F

F

{

T F F

tr Tr T

1.

(p)q)<=(-p V

Ja nette does her old car.

q) and -( p)q)<=+(pn-q) dcrtl Y:r.ltr ^[w co-r: t1 s< tlr ot<J.-

not bdy a riew car or she sells'9d (Lv^ V C

n-) o

1.

It is false that WorkinS.

if Lam=told then the 11, Ca-€'u-i) C n '{^,-'UU

\".J

heater is

Name

Variations of the Conditional Statement Symbolic form

Read

Conditional

Converse of the conditional Inverse of the conditional Contrapositive of the conditional

p'-+ q

qlp

*P--'+*cl -q-*p

"lf p, then qo'

'"lf 4. then p"

"If not p, then not {" "lf not q, then not p"

Write the converse, inverse and

contrapositive for:

lf I go to Ita[y then I buy a leather bag. 'L -1 \2

C: b *--? i

l: n, \.-:n

CP: \- \a '* ) 4= |

True or false? lf a natural number is not divisible by 7 then a natural number is not divisible by 14.

t o\ nclt}b \5 chutstbu h9

rtt

Contra positive:\q(5.T{LUZ

\

thtn r$ C\ruttrhtl h\,,5-t

q