0.3 Discrete structures logic

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Logic

Logic is a language for reasoning. It is a collection of rules we use when doing logical reasoning. Human reasoning has been observed over centuries from at least the times of Greeks, and patterns appearing in reasoning have been extracted, abstracted, and streamlined. The foundation of the logic we are going to learn here was laid down by a British mathematician George Boole in the middle of the 19th century, and it was further developed and used in an attempt to derive all of mathematics by Gottlob Frege, a German mathematician, towards the end of the 19th century. A British philosopher/mathematician, Bertrand Russell, found a flaw in basic assumptions in Frege's attempt but he, together with Alfred Whitehead, developed Frege's work further and repaired the damage. The logic we study today is more or less along this line.

In logic we are interested in true or false of statements, and how the truth/falsehood of a statement can be determined from other statements. However, instead of dealing with individual specific statements, we are going to use symbols to represent arbitrary statements so that the results can be used in many similar but different cases. The formalization also promotes the clarity of thought and eliminates mistakes.

There are various types of logic such as logic of sentences (propositional logic), logic of objects (predicate logic), logic involving uncertainties, logic dealing with fuzziness, temporal logic etc. Here we are going to be concerned with propositional logic and predicate logic, which are fundamental to all types of logic.

Introduction to propositional logic

Propositional logic is a logic at the sentential level. The smallest unit we deal with in propositional logic is a sentence. We do not go inside individual sentences and analyze or discuss their meanings. We are going to be interested only in true or false of sentences, and major concern is whether or not the truth or falsehood of a certain sentence follows from those of a set of sentences, and if so, how. Thus sentences considered in this logic are not arbitrary sentences but are the ones that are true or false. This kind of sentences are called propositions.

What is proposition?

Sentences considered in propositional logic are not arbitrary sentences but are the ones that are either true or false, but not both. This kind of sentences are called propositions. If a proposition is true, then we say it has a truth value of "true"; if a proposition is false, its truth value is "false".

For example, "Grass is green", and "2 + 5 = 5" are propositions. The first proposition has the truth value of "true" and the second "false".

But "Close the door", and "Is it hot outside?" are not propositions. Also "x is greater than 2", where x is a variable representing a number, is not a proposition, because unless a specific value is given to x we can not say whether it is true or false, nor do we know what x represents.

Similarly "x = x" is not a proposition because we don't know what "x" represents hence what "=" means. For example, while we understand what "3 = 3" means, what does "Air is equal to air" or "Water is equal to water" mean? Does it mean a mass of air is equal to another mass or the concept of air is equal to the concept of air? We don't quite know what "x = x" mean. Thus we can not say whether it is true or not. Hence it is not a proposition.

List and explain four factors of production
capital labour entrepreneur natural resources
Thembi
What is supply
when the supply decreases demand also decreases
Thembi
types of demand and the explanation
what is demand
other things remaining same if demend is increases supply is also decrease and if demend is decrease supply is also increases is called the demand
Mian
if the demand increase supply also increases
Thembi
you are wrong this is the law of demand and not the definition
Tarasum
Demand is the willingness of buy and ability to buy in a specific time period in specific place. Mian you are saying law of demand but not in proper way. you have to keep studying more. because its very basic things in Economics.
Hamza
Demand is the price of Quantity goods and services in which consumer's are willing and able to offer at a price in the market over a period of time
Umar
Demand is the quantity of goods and services that the consumer are willing and able to buy at a alternative prices over a given period of time. But mind you demand is quite different from need and want.
Tarasum
Demand can be defined as the graphical representation between price&demand
alkasim
sorry demand is nt a graphical representation between price and quantity demand but instead that is demand curve.
Ebrima
Demand is the willingness and ability of a consumer to buy a quantity of a good over a given period of time assuming all other things remain constant.
Vedaant
what is commercialization?
How to talk loan for bank?
what is the meaning of gpa?
Answer: GPA stands for Grade Point Average. It is a standard way of measuring academic achievement in the U.S. Basically, it goes as follows: Each course is given a certain number of "units" or "credits", depending on the content of the course.
Yusuf
what is small and Microbuisenes
What is fiscal policy
Dansofo
Who is the funder of Economic
Dansofo
founder , that is Adam Smith
Daniel
what is model
The wealth of Nations
the wealth of nations, is it the first?
Umar
Yes very sure it was released in 1759
Yusuf
thank you Yusuf.
Umar
then when did he died?
Umar
17 July 1790 Born: 16 June 1723, Kirkcaldy, United Kingdom Place of death: Panmure House, Edinburgh, United Kingdom
Yusuf
1790
Yusuf
that's my today questions, thank you Yusuf it's bed time see u after.
Umar
what is fiscal policy
what's mode?
mode is the highest occurring frequency in a distribution
Bola
mode is the most commonly occurring item in a set of data.
Umar
Please, what is the difference between monopoly and monopsony?
is there monopsony word?
Umar
I have no idea though
Umar
please, in which year Adam smith was born?
Umar
monopsony is when there's only one buyer while monopoly is when there's only one producer.
Bola
who have idea on Banter
Ibrahim
like trade by barter?
Bola
Monopoly is when there's excessively one seller and there is no entry in the market while monopsony is when there is one buyer
kemigisha
Adam smith was born in 1723
Bola
(uncountable) Good humoured, playful, typically spontaneous conversation. verb (intransitive) To engage in banter or playful conversation. (intransitive) To play or do something amusing. (transitive) To tease mildly.
Umar
which book Adam smith published first? the first book of Adam smith pls.
Umar
wealth on nation, 1776
Daniel
what is market power and how can it affect an economy?
market power:- where a firm is said to be a price setter.market power benefits the powerful at the expense of others.
Umar
Market power refers to the ability of a firm (or group of firms) to raise and maintain price above the level that would prevail under competition is referred to as market or monopoly power. The exercise of market power leads to reduced output and loss of economic welfare
Kartheek
find information about the national budget
Molahlegi
three branches of economics in which tourism is likely to figure
What are those three branches?
IlRegno
While the American heart association suggests that meditation might be used in conjunction with more traditional treatments as a way to manage hypertension
in a comparison of the stages of meiosis to the stage of mitosis, which stages are unique to meiosis and which stages have the same event in botg meiosis and mitosis
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