# 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).

are nano particles real
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
no can't
Lohitha
where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
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
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
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
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
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
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?
what is a peer
What is meant by 'nano scale'?
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 ?
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?
what king of growth are you checking .?
Renato
While the American heart association suggests that meditation might be used in conjunction with more traditional treatments as a way to manage hypertension
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