<< Chapter < Page Chapter >> Page >

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

the diagram of the digestive system
Assiatu Reply
How does twins formed
William Reply
They formed in two ways first when one sperm and one egg are splited by mitosis or two sperm and two eggs join together
Oluwatobi
what is genetics
Josephine Reply
Genetics is the study of heredity
Misack
how does twins formed?
Misack
What is manual
Hassan Reply
discuss biological phenomenon and provide pieces of evidence to show that it was responsible for the formation of eukaryotic organelles
Joseph Reply
what is biology
Yousuf Reply
the study of living organisms and their interactions with one another and their environments
AI-Robot
the study of living organisms and their interactions with one another and their environment.
Wine
discuss the biological phenomenon and provide pieces of evidence to show that it was responsible for the formation of eukaryotic organelles in an essay form
Joseph Reply
what is the blood cells
Shaker Reply
list any five characteristics of the blood cells
Shaker
lack electricity and its more savely than electronic microscope because its naturally by using of light
Abdullahi Reply
advantage of electronic microscope is easily and clearly while disadvantage is dangerous because its electronic. advantage of light microscope is savely and naturally by sun while disadvantage is not easily,means its not sharp and not clear
Abdullahi
cell theory state that every organisms composed of one or more cell,cell is the basic unit of life
Abdullahi
is like gone fail us
DENG
cells is the basic structure and functions of all living things
Ramadan
What is classification
ISCONT Reply
is organisms that are similar into groups called tara
Yamosa
in what situation (s) would be the use of a scanning electron microscope be ideal and why?
Kenna Reply
A scanning electron microscope (SEM) is ideal for situations requiring high-resolution imaging of surfaces. It is commonly used in materials science, biology, and geology to examine the topography and composition of samples at a nanoscale level. SEM is particularly useful for studying fine details,
Hilary
cell is the building block of life.
Condoleezza Reply
what is cell divisoin?
Aron Reply
Diversity of living thing
ISCONT
what is cell division
Aron Reply
Cell division is the process by which a single cell divides into two or more daughter cells. It is a fundamental process in all living organisms and is essential for growth, development, and reproduction. Cell division can occur through either mitosis or meiosis.
AI-Robot
What is life?
Allison Reply
life is defined as any system capable of performing functions such as eating, metabolizing,excreting,breathing,moving,Growing,reproducing,and responding to external stimuli.
Mohamed
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Discrete structures. OpenStax CNX. Jan 23, 2008 Download for free at http://cnx.org/content/col10513/1.1
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Discrete structures' conversation and receive update notifications?

Ask