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For example, specifying the set {1, 3, 5} as the universe and assigning 0 to the variable y, for example, is an interpretation for the wff ∀x Q(x, y), where Q(x, y) means x is greater than y. ∀x Q(x, y) with that interpretation reads, for example, "Every number in the set {1, 3, 5} is greater than 0".

As can be seen from the above example, a wff becomes a proposition when it is given an interpretation.

There are, however, wffs which are always true or always false under any interpretation. Those and related concepts are discussed below.

Satisfiable, unsatisfiable and valid wffs

A wff is said to be satisfiable if there exists an interpretation that makes it true, that is if there are a universe, specific predicates assigned to the predicate variables, and an assignment of values to the free variables that make the wff true.

For example, ∀x N(x), where N(x) means that x is non-negative, is satisfiable. For if the universe is the set of natural numbers, the assertion ∀x N(x) is true, because all natural numbers are non-negative. Similarly ∃x N(x) is also satisfiable.

However, ∀x [N(x) ⋀¬N(x)] is not satisfiable because it can never be true. A wff is called invalid or unsatisfiable, if there is no interpretation that makes it true.

A wff is valid if it is true for every interpretation*. For example, the wff ∀x P(x) ⋁∃x ¬P(x) is valid for any predicate name P , because ∃x ¬P(x) is the negation of ∀x P(x). However, the wff ∀x N(x) is satisfiable but not valid.

Note that a wff is not valid iff it is unsatisfiable for a valid wff is equivalent to true. Hence its negation is false.

Equivalence

Two wffs W1 and W2 are equivalent if and only if W1 ↔W2 is valid, that is if and only if W1 ↔W2 is true for all interpretations.

For example ∀x P(x) and ¬∃x ¬P(x) are equivalent for any predicate name P. So are ∀x [ P(x) ⋀Q(x) ] and [ ∀x P(x) ⋀∀x Q(x) ]for any predicate names P and Q .

Transcribing english to predicate logic wffs

English sentences appearing in logical reasoning can be expressed as a wff. This makes the expressions compact and precise. It thus eliminates possibilities of misinterpretation of sentences. The use of symbolic logic also makes reasoning formal and mechanical, contributing to the simplification of the reasoning and making it less prone to errors.

Transcribing English sentences into wffs is sometimes a non-trivial task. In this course we are concerned with the transcription using given predicate symbols and the universe. To transcribe a proposition stated in English using a given set of predicate symbols, first restate in English the proposition using the predicates, connectives, and quantifiers. Then replace the English phrases with the corresponding symbols.

Example: Given the sentence "Not every integer is even", the predicate "E(x)" meaning x is even, and that the universe is the set of integers, first restate it as "It is not the case that every integer is even" or "It is not the case that for every object x in the universe, x is even."

Then "it is not the case" can be represented by the connective "¬", "every object x in the universe" by "∀ x", and "x is even" by E(x).

Questions & Answers

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
Do somebody tell me a best nano engineering book for beginners?
s. Reply
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
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
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
Prasenjit Reply
in a comparison of the stages of meiosis to the stage of mitosis, which stages are unique to meiosis and which stages have the same event in botg meiosis and mitosis
Leah 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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