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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

how can chip be made from sand
Eke Reply
is this allso about nanoscale material
Almas
are nano particles real
Missy Reply
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
Lale Reply
no can't
Lohitha
where is the latest information on a no technology how can I find it
William
currently
William
where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
Bhagvanji
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
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
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
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
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
While the American heart association suggests that meditation might be used in conjunction with more traditional treatments as a way to manage hypertension
Beverly 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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