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3. ∃x [ P(x) ⋁Q(x) ] ⇔[ ∃x P(x) ⋁∃x Q(x) ], again for the same example, can be shown in Figure 3:

LHS says someone is rich or happy, and RHS says someone is rich or someone is happy. Thus clearly LHS implies RHS. Also if someone is rich then that person is certainly rich or happy. Thus RHS implies LHS.

4. ∃x [ P(x) ⋀Q(x) ] ⇒[ ∃x P(x) ⋀∃x Q(x) ], for the same example, can be shown in Figure 4:

LHS say someone is rich and happy. Hence there is someone who is rich and there is someone who is happy. Hence LHS implies RHS. However, since RHS can be true without anyone being rich and happy at the same time, RHS does not necessarily imply LHS.

Quantifiers and connectives 2

If a wff (Q below) in the scope of a quantifier does not have the variable (x below) that is quantified by that quantifier, then that wff can be taken out of the scope of that quantifier. That is,

1. ∀x [ P(x) ⋀Q ] ⇔[ ∀x P(x) ⋀Q ]

2. [ ∀x P(x) ⋁Q ] ⇔∀x [ P(x) ⋁Q ]

3. ∃x [ P(x) ⋁Q ] ⇔[ ∃x P(x) ⋁Q ]

4. ∃x [ P(x) ⋀Q ] ⇔[ ∃x P(x) ⋀Q ],

where Q in all these formulas DO NOT have the variable x .

Note: When implication → and/or equivalence ↔ are involved, you can not necessarily take Q outside the scope. To see what happens, express → and ↔ using ⋁and ⋀, and apply the above formulas. For example ∀x [ P(x)→Q ] is NOT equivalent to ∀x P(x)→Q . Rather it is equivalent to ∃x P(x)→Q. Further details are left as an exercise.

Questions and exercises

1. Which of the following sentences is a proposition?

a. Every one is happy.

b. If it snows, then schools are closed in Norfolk, VA.

c. x + 2 is positive

d. Take an umbrella with you.

e. I suggest that you take an umbrella with you

2. Which of the following tables is a truth table?

Z below represents a proposition involving P and Q.

Table 1
P Q Proposition Z
F F F
T F T
T T T
T F T
Table 2
P Q Proposition Z
F F F
T F T
T T F
Table 3
P Q Proposition Z
F F F
F T T
T F T
T T T
Table 4
P Proposition Z
F F
F T
T F

3. Indicate which of the following statements are correct and which ones are incorrect.

a. If P is True and Q is False, then P⋀Q is True.

b. If P is False and Q is True, then P → Q is True.

c. If P is False and Q is False, then P ↔ Q is False

d. If P is True and Q is False, then P ⋁ Q is True.

e. If P is True and Q is False, then ¬[P⋀Q] is False

4. Indicate which of the following expressions are propositions and which are not.

a. P⋀¬Q.

b. [[P ⋁ Q] → [Q ⋀ R]]

c. [¬[P ↔ ⋀ Q ] ⋁ Q ]

d. [¬¬P ⋁ Q]

e. [[Q ⋁ R][P ⋀ Q]]

5. Indicate which of the following converses and contrapositives are correct and which are not.

a. If it snows, the schools will be closed.

Converse: If the schools are closed, it snows.

Contrapositive: If the schools are not closed, it does not snow.

b. If I work all night, I can finish this project.

Converse: If I cannot finish this project, I work all night.

Contrapositive: If I can finish this project, I don’t work all night.

c. I eat spicy food, only if it upsets my stomach.

Converse: If I eat spicy food, it upsets my stomach.

Contrapositive: If I don’t eat spicy food, it doesn’t upset my stomach.

6. Which of the following pairs of propositions are logically equivalent?

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Discrete structures. OpenStax CNX. Jan 23, 2008 Download for free at http://cnx.org/content/col10513/1.1
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