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To prove this, first let us get rid of → using one of the identities: (P→Q ) ⇔( ¬P ⋁Q).

That is, ¬( P →Q ) ⇔¬( ¬P ⋁Q ).

Then by De Morgan, it is equivalent to ¬¬P ⋀¬Q , which is equivalent to P ⋀¬Q, since the double negation of a proposition is equivalent to the original proposition as seen in the identities.

2. P ⋁( P ⋀Q ) ⇔P --- Absorption

What this tells us is that P ⋁( P ⋀Q ) can be simplified to P, or if necessary P can be expanded into P ⋁( P ⋀Q ) .

To prove this, first note that P ⇔( P ⋀T ).

Hence

P ⋁( P ⋀Q )

⇔( P ⋀T ) ⋁( P ⋀Q )

⇔P ⋀( T ⋁Q ) , by the distributive law.

⇔( P ⋀T ) , since ( T ⋁Q ) ⇔T.

⇔P , since ( P ⋀T ) ⇔P.

Note that by the duality

P ⋀( P ⋁Q ) ⇔P also holds.

Implications

The following implications are some of the relationships between propositions that can be derived from the definitions (meaning) of connectives. ⇒ below corresponds to → and it means that the implication always holds. That is it is a tautology.

These implications are used in logical reasoning. When the right hand side of these implications is substituted for the left hand side appearing in a proposition, the resulting proposition is implied by the original proposition, that is, one can deduce the new proposition from the original one.

First the implications are listed, then examples to illustrate them are given. List of Implications:

1. P ⇒(P ⋁Q) ----- addition

2. (P ⋀Q) ⇒P ----- simplification

3. [P ⋀(P →Q] ⇒Q ----- modus ponens

4. [(P →Q) ⋀¬Q] ⇒¬P ----- modus tollens

5. [ ¬P ⋀(P ⋁Q] ⇒Q ----- disjunctive syllogism

6. [(P →Q) ⋀(Q→R)] ⇒(P→R) ----- hypothetical syllogism

7. (P→Q) ⇒[(Q→R)→(P→R)]

8. [(P→Q) ⋀(R→S)] ⇒[(P ⋀R)→(Q ⋀S)]

9. [(P ↔Q) ⋀(Q ↔R)] ⇒(P ↔R)

Examples:

1. P ⇒(P ⋁Q) ----- addition

For example, if the sun is shining, then certainly the sun is shining or it is snowing. Thus

"if the sun is shining, then the sun is shining or it is snowing." "If 0<1, then 0 ≤1 or a similar statement is also often seen.

2. (P ⋀Q) ⇒P ----- simplification

For example, if it is freezing and (it is) snowing, then certainly it is freezing. Thus "If it is freezing and (it is) snowing, then it is freezing."

3. [P ⋀(P →Q] ⇒Q ----- modus ponens

For example, if the statement "If it snows, the schools are closed" is true and it actually snows, then the schools are closed.

This implication is the basis of all reasoning. Theoretically, this is all that is necessary for reasoning. But reasoning using only this becomes very tedious.

4. [(P →Q) ⋀¬Q] ⇒¬P ----- modus tollens

For example, if the statement "If it snows, the schools are closed" is true and the schools are not closed, then one can conclude that it is not snowing. Note that this can also be looked at as the application of the contrapositive and modus ponens. That is, (P→Q) is equivalent to ( ¬Q )→( ¬P ). Thus if in addition ¬Q holds, then by the modus ponens, ¬P is concluded.

5. [ ¬P ⋀(P ⋁Q] ⇒Q ----- disjunctive syllogism

For example, if the statement "It snows or (it) rains." is true and it does not snow, then one can conclude that it rains.

6. [(P→Q) ⋀(Q→R)] ⇒(P→R) ----- hypothetical syllogism

Questions & Answers

can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
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
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
Commplementary angles
Idrissa Reply
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.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
kkk nice
Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
is it 3×y ?
Joan Reply
J, combine like terms 7x-4y
Bridget Reply
im not good at math so would this help me
Rachael Reply
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
linda Reply
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris 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
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
can nanotechnology change the direction of the face of the world
Prasenjit Reply
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.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
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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