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Example of inferencing

Consider the following argument:

1. Today is Tuesday or Wednesday.

2. But it can't be Wednesday, since the doctor's office is open today, and that office is always closed on Wednesdays.

3. Therefore today must be Tuesday.

This sequence of reasoning (inferencing) can be represented as a series of application of modus ponens to the corresponding propositions as follows.

The modus ponens is an inference rule which deduces Q from P → Q and P.

T: Today is Tuesday.

W: Today is Wednesday.

D: The doctor's office is open today.

C: The doctor's office is always closed on Wednesdays.

The above reasoning can be represented by propositions as follows.

1. T ⋁ W

2. D

C

------------

~W

------------

3. T

To see if this conclusion T is correct, let us first find the relationship among C, D, and W:

C can be expressed using D and W. That is, restate C first as the doctor's office is always closed if it is Wednesday. Then C ↔ (W → ~D) Thus substituting (W → ~D) for C, we can proceed as follows.

D

W → ~D

------------

~W

which is correct by modus tollens.

From this ~W combined with T V W of 1. above,

~W

T ⋁ W

------------

T

which is correct by disjunctive syllogism.

Thus we can conclude that the given argument is correct.

To save space we also write this process as follows eliminating one of the ~W's:

D

W → ~D

------------

~W

T ⋁ W

------------

T

Proof of identities

All the identities in Section Identities can be proven to hold using truth tables as follows. In general two propositions are logically equivalent if they take the same value for each set of values of their variables. Thus to see whether or not two propositions are equivalent, we construct truth tables for them and compare to see whether or not they take the same value for each set of values of their variables.

For example consider the commutativity of ⋁:

(P ⋁Q) ⇔(Q ⋁P).

To prove that this equivalence holds, let us construct a truth table for each of the proposition (P ⋁Q) and (Q ⋁P).

A truth table for (P ⋁Q) is, by the definition of ⋁,

P Q (P ⋁Q)
F F F
F T T
T F T
T T T

A truth table for (Q ⋁P) is, by the definition of ⋁,

P Q (Q ⋁P)
F F F
F T T
T F T
T T T

As we can see from these tables (P ⋁Q) and (Q ⋁P) take the same value for the same set of value of P and Q. Thus they are (logically) equivalent.

We can also put these two tables into one as follows:

P Q (P ⋁Q) (Q ⋁P)
F F F F
F T T T
T F T T
T T T T

Using this convention for truth table we can show that the first of De Morgan's Laws also holds.

P Q ¬(P ⋁Q) ¬P ⋀¬Q
F F T T
F T F F
T F F F
T T F F

By comparing the two right columns we can see that ¬(P ⋁Q) and ¬P ⋀¬Q are equivalent.

Proof of implications

1. All the implications in Section Implications can be proven to hold by constructing truth tables and showing that they are always true.

For example consider the first implication "addition": P ⇒ (P ⋁ Q).

To prove that this implication holds, let us first construct a truth table for the proposition P ⋁ Q.

P Q (P ⋁ Q)
F F F
F T T
T F T
T T T

Then by the definition of →, we can add a column for P → (P ⋁ Q) to obtain the following truth table.

P Q (P ⋁ Q) P →(P ⋁ Q)
F F F T
F T T T
T F T T
T T T T

Questions & Answers

what is the stm
Brian Reply
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?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
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
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
why?
Adin
what school?
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
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
for teaching engĺish at school how nano technology help us
Anassong
How can I make nanorobot?
Lily
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
how can I make nanorobot?
Lily
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
Difference between extinct and extici spicies
Amanpreet 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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