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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

Don't forget about the age old question of What is the difference between mala in se and mala prohibita?

If you want to learn more check out Where is olmec located?

Negation: (-) We also discuss several other topics like What is the difference between pulmonary pump and systemic pump?

Texas is nr* the largest state.(^p )T 1,. If you want to learn more check out What are the ways to generate negative effect?

Oklahoma c\urr nr\ border Texas. (^ .1 )tr 2.

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) Don't forget about the age old question of Why does this particular behavior persist in organisms?

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

All are .1. Some are

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. We also discuss several other topics like Who used an atomizer to spray oil drop particles in a drum with air and then calculate the charge on an electron by calculating the mass of one?

Let r = The roads are icy.

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<

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#q

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

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