# 0.3 Discrete structures logic  (Page 3/23)

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In everyday life we often combine propositions to form more complex propositions without paying much attention to them. For example combining "Grass is green", and "The sun is red" we say something like "Grass is green and the sun is red", "If the sun is red, grass is green", "The sun is red and the grass is not green" etc. Here "Grass is green", and "The sun is red" are propositions, and form them using connectives "and", "if... then ..." and "not" a little more complex propositions are formed. These new propositions can in turn be combined with other propositions to construct more complex propositions. They then can be combined to form even more complex propositions. This process of obtaining more and more complex propositions can be described more generally as follows:

Let X and Y represent arbitrary propositions. Then [¬X], [X⋀Y], [X⋁Y], [X→Y], and [X↔Y] are propositions.

Note that X and Y here represent an arbitrary proposition. This is actually a part of more rigorous definition of proposition which we see later.

Example : [ P → [Q ⋁ R] ]is a proposition and it is obtained by first constructing [Q ⋁ R] by applying [X ⋁ Y]to propositions Q and R considering them as X and Y, respectively, then by applying [ X→Y ] to the two propositions P and [Q ⋁ R]considering them as X and Y, respectively.

Note: Rigorously speaking X and Y above are place holders for propositions, and so they are not exactly a proposition. They are called a propositional variable, and propositions formed from them using connectives are called a propositional form. However, we are not going to distinguish them here, and both specific propositions such as "2 is greater than 1" and propositional forms such as (P ⋁Q) are going to be called a proposition.

## Converse and contrapositive

For the proposition P→Q, the proposition Q→P is called its converse, and the proposition ¬ Q→ ¬ P is called its contrapositive.

For example for the proposition "If it rains, then I get wet",

Converse: If I get wet, then it rains.

Contrapositive: If I don't get wet, then it does not rain.

The converse of a proposition is not necessarily logically equivalent to it, that is they may or may not take the same truth value at the same time.

On the other hand, the contrapositive of a proposition is always logically equivalent to the proposition. That is, they take the same truth value regardless of the values of their constituent variables. Therefore, "If it rains, then I get wet." and "If I don't get wet, then it does not rain." are logically equivalent. If one is true then the other is also true, and vice versa.

## If_then variations

If-then statements appear in various forms in practice. The following list presents some of the variations. These are all logically equivalent, that is as far as true or false of statement is concerned there is no difference between them. Thus if one is true then all the others are also true, and if one is false all the others are false.

• If p, then q.
• p implies q.
• If p, q.
• p only if q.
• p is sufficient for q.
• q if p.
• q whenever p.
• q is necessary for p.
• It is necessary for p that q.

#### Questions & Answers

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
Daniel
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
characteristics of micro business
Abigail
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
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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