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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

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.
yes that's correct
I think
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
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
analytical skills graphene is prepared to kill any type viruses .
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
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
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
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
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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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