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Similarly the existential quantifier turns, for example, the statement x>1 to "for some object x in the universe, x>1", which is expressed as "∃x x>1." Again, it is true or false in the universe of discourse, and hence it is a proposition once the universe is specified.

Universe of discourse

The universe of discourse, also called universe, is the set of objects of interest. The propositions in the predicate logic are statements on objects of a universe. The universe is thus the domain of the (individual) variables. It can be the set of real numbers, the set of integers, the set of all cars on a parking lot, the set of all students in a classroom etc. The universe is often left implicit in practice. But it should be obvious from the context.

The universal quantifier

The expression: ∀x P(x), denotes the universal quantification of the atomic formula P(x). Translated into the English language, the expression is understood as: "For all x, P(x) holds", "for each x, P(x) holds" or "for every x, P(x) holds". ∀is called the universal quantifier, and ∀x means all the objects x in the universe. If this is followed by P(x) then the meaning is that P(x) is true for every object x in the universe. For example, "All cars have wheels" could be transformed into the propositional form, ∀x P(x), where:

  • P(x) is the predicate denoting: x has wheels, and
  • the universe of discourse is only populated by cars.

Universal quantifier and connective and

If all the elements in the universe of discourse can be listed then the universal quantification ∀x P(x) is equivalent to the conjunction: P(x1)) ⋀P(x2) ⋀P(x3) ⋀... ⋀P(xn) . For example, in the above example of ∀x P(x), if we knew that there were only 4 cars in our universe of discourse (c1, c2, c3 and c4) then we could also translate the statement as: P(c1) ⋀P(c2) ⋀P(c3) ⋀P(c4) .

The existential quantifier

The expression: ∃xP(x), denotes the existential quantification of P(x). Translated into the English language, the expression could also be understood as: "There exists an x such that P(x)" or "There is at least one x such that P(x)" ∃ is called the existential quantifier, and ∃x means at least one object x in the universe. If this is followed by P(x) then the meaning is that P(x) is true for at least one object x of the universe. For example, "Someone loves you" could be transformed into the propositional form, ∃x P(x), where:

  • P(x) is the predicate meaning: x loves you,
  • The universe of discourse contains (but is not limited to) all living creatures.

Existential quantifier and connective or

If all the elements in the universe of discourse can be listed, then the existential quantification ∃xP(x) is equivalent to the disjunction: P(x1) ⋁P(x2) ⋁P(x3) ⋁... ⋁P(xn).

For example, in the above example of ∃x P(x), if we knew that there were only 5 living creatures in our universe of discourse (say: me, he, she, rex and fluff), then we could also write the statement as: P(me)⋁P(he) ⋁P(she) ⋁P(rex) ⋁P(fluff)

An appearance of a variable in a wff is said to be bound if either a specific value is assigned to it or it is quantified. If an appearance of a variable is not bound, it is called free. The extent of the application (effect) of a quantifier, called the scope of the quantifier, is indicated by square brackets [ ]. If there are no square brackets, then the scope is understood to be the smallest wff following the quantification.

Questions & Answers

anyone know any internet site where one can find nanotechnology papers?
Damian Reply
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.
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
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
characteristics of micro business
for teaching engĺish at school how nano technology help us
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
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
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.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
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.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
many many of nanotubes
what is the k.e before it land
what is the function of carbon nanotubes?
I'm interested in nanotube
what is nanomaterials​ and their applications of sensors.
Ramkumar 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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