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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

Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
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
hi
Loga
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
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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