# 0.3 Discrete structures logic  (Page 16/23)

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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

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
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Adin
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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
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
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Damian Reply
absolutely yes
Daniel
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Maciej
characteristics of micro business
Abigail
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Anassong
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
NANO
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
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
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
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Cied
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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