# 0.3 Discrete structures logic  (Page 10/23)

 Page 10 / 23

## 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

can someone help me with some logarithmic and exponential equations.
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
Commplementary angles
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
a perfect square v²+2v+_
kkk nice
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
rolling four fair dice and getting an even number an all four dice
Kristine 2*2*2=8
Differences Between Laspeyres and Paasche Indices
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
is it 3×y ?
J, combine like terms 7x-4y
im not good at math so would this help me
yes
Asali
I'm not good at math so would you help me
Samantha
what is the problem that i will help you to self with?
Asali
how do you translate this in Algebraic Expressions
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
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
what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
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
can nanotechnology change the direction of the face of the world
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!