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About “A Trip To Mathematics”:
A Trip to Mathematics is an indefinitely long series, aimed on generally interested readers and other undergraduate students. This series will deal Basic Mathematics as well as Advanced Mathematics in very interactive manners. Each post of this series is kept small that reader be able to grasp concepts. Critics and suggestions are invited in form of comments.
What is Logic?
If mathematics is regarded as a language, then logic is its grammar.
In other words, logical precision has the same importance in mathematics as grammatical accuracy in a language. As linguistic grammar has sentences, statements— logic has them too. Let we discuss about Sentence &Statements, then we shall proceed to further logic .
Sentence & Statements
A sentence is a collection of some words, those together having some sense.
- Math is a tough subject.
- English is not a tough subject.
- Math and English both are tough subjects.
- Either Math or English is tough subject.
- If Math is a tough subject, then English is also a tough subject.
- Math is a tough subject, if and only if English is a tough subject.
Just have a quick look on above collections of words. Those are sentences, since they have some meaning too. First sentence is called Prime Sentence, i.e., sentence which either contains no connectives or, by choice, is regarded as “indivisible”. The five words
- if …. then
- if and only if
or their combinations are called ‘connectives‘. The sentences (all but first) are called composite sentences, i.e., a declarative sentence (statement) in which one or more connectives appear. Remember that there is no difference between a sentence and statement in general logic. In this series, sentences and statements would have the same meaning.
not: A sentence which is modified by the word “not” is called the negation of the original sentence.
For example: “English is not a tough subject” is the negation of “English is a tough subject“. Also, “3 is not a prime” is the negation of “3 is a prime“. Always note that negation doesn’t really mean the converse of a sentence. For example, you can not write “English is a simple subject” as the negation of “English is a tough subject“.
In mathematical writings, symbols are often used for conciseness. The negation of sentences/statements is expressed by putting a slash (/) over that symbol which incorporates the principal verb in the statement.
For example: The statement (read ‘x is equal to y’) is negated as (read ‘x is not equal to y‘). Similarly, (read ‘x does not belong to set A‘) is the negation of (read ‘x belongs to set A‘).
Statements are sometimes represented by symbols like p, q, r, s etc. With this notation there is a symbol, or ¬ (read as ‘not’) for negation. For example if ‘p’ stands for the statement “Terence Tao is a professor” then [or ¬p] is read as ‘not p’ and states for “Terence Tao is not a professor.” Sometimes ~p is also used for the negation of p.
and: The word “and” is used to join two sentences to form a composite sentence which is called the conjunction of the two sentences. For example, the sentence “I am writing, and my sister is reading” is the conjunction of the two sentences: “I am writing” and “My sister is reading“. In ordinary language (English), words like “but, while” are used as approximate synonyms for “and“, however in math, we shall ignore possible differences in shades of meaning which might accompany the use of one in the place of the other. This allows us to write “I am writing but my sister is reading” having the same mathematical meaning as above.
The standard notation for conjunction is , read as ‘and‘. If p and q are statements then their conjunction is denoted by and is read as ‘p and q’.
or: A sentence formed by connecting two sentences with the word “or” is called the disjunction of the two sentences. For example, “Justin Bieber is a celebrity, or Sachin Tendulkar is a footballer.” is a disjunction of “Justin Bieber is a celebrity” and “Sachin Tendulkar is a footballer“.
Sometimes we put the word ‘either‘ before the first statement to make the disjunction sound nice, but it is not necessary to do so, so far as a logician is concerned. The symbolic notation for disjunction is read ‘or’. If p and q are two statements, their disjunction is represented by and read as p or q.
if….then: From two sentences we may construct one of the from “If . . . . . then . . .“; which is called a conditional sentence. The sentence immediately following IF is the antecedent, and the sentence immediately following THEN is the consequent. For example, “If 5 <6, 6<7, then 5<7” is a conditional sentence whith “5<6, 6<7” as antecedent and “5<7” as consequent. If p and q are antecedent and consequent sentences respectively, then the conditional sentence can be written as:
“If p then q”.
This can be mathematically represented as and is read as “p implies q” and the statement sometimes is also called implication statement. Several other ways are available to paraphrase implication statements including:
- If p then q
- p implies q
- q follows from p
- q is a logical consequence of p
- p (is true) only if q (is true)
- p is a sufficient condition for q
- q is a necessary condition for p
If and Only If: The phrase “if and only if” (abbreviated as ‘iff‘) is used to obtain a bi-conditional sentence. For example, “A triangle is called a right angled triangle, if and only if one of its angles is 90°.” This sentence can be understood in either ways: “A triangle is called a right angled triangle if one of its angles is 90°” and “One of angles of a triangle is 90° if the triangle is right angled triangle.” This means that first prime sentence implies second prime sentence and second prime sentence implies first one. (This is why ‘iff’ is sometimes called double-implication.)
Another example is “A glass is half filled iff that glass is half empty.“
If p and q are two statements, then we regard the biconditional statement as “p if and only if q” or “p iff q” and mathematically represent by ” “. represents double implication and read as ‘if and only if’.
In the statement , the implication is called direct implication and the implication is called the converse implication of the statement.
Other terms in logic:
Stronger and Weaker Statements: A statement p is stronger than a statement q (or that q is weaker than p) if the implication statement is true.
Strictly Stronger and Strictly Weaker Statements: The word ‘stronger‘(or weaker) does not necessarily mean ‘strictly stronger‘ (or strictly weaker).
For example, every statement is stronger than itself, since . The apparent paradox here is purely linguistical. If we want to avoid it, we should replace the word stronger by the phrase ‘stronger than‘ or ‘possibly as strong as‘.
If is true but its converse is false (), then we say that p is strictly stronger than q (or that q is strictly weaker than p). For example it is easy to say that a given quadrilateral is a rhombus that to say it is a parallelogram. Another understandable example is that ” If a blog is hosted on WordPress.com, it is powered with WordPress software.” is true but ” If a blog is powered with WordPress software , it is hosted on WordPress.com” is not true.
What exactly is the difference between a mathematician, a physicist and a layman?
Let us suppose they all start measuring the angles of hundreds of triangles of various shapes, find the sum in each case and keep a record.
Suppose the layman finds that with one or two exceptions the sum in each case comes out to be 180 degrees. He will ignore the exceptions and state ‘The sum of the three angles in a triangle is 180 degrees.’
A physicist will be more cautious in dealing the exceptional cases. He will examine then more carefully. If he finds that the sum in them some where 179 degrees to 181 degrees, say, then if will attribute the deviation to experimental errors. He will state a law – ‘The sum of the three angles of any triangle is 180 degrees.’ He will then watch happily as the rest of the world puts his law to test and finds that it holds good in thousands of different cases, until somebody comes up with a triangle in which the law fails miserably. The physicist now has to withdraw his law altogether or else to replace it by some other law which holds good in all the cases tried. Even this new law may have to be modified at a later date. And this will continue without end.
A mathematician will be the fussiest of all. If there is even a single exception, he will refrain from saying anything. Even when millions of triangles are tried without a single exception, he will not state it as a theorem that the sum of the three angles in ‘any’ triangle is 180 degrees. The reason is that there are infinitely many different types of triangles. To generalise from a million to infinity is as baseless to a mathematician as to generalise from one to a million. He will at the most make a conjecture and say that there is a ‘strong evidence’ suggesting that the conjecture is true.
The approach taken by the layman or the physicist is known as the inductive approach whereas the mathematician’s approach is called the deductive approach.
In inductive approach, we make a few observations and generalise. Exceptions are generally not counted in inductive approach.
In this approach, we deduce from something which is already proven.
Axioms or Postulates
Sometimes, when deducting theorems or conclusion from another theorems, we reach at a stage where a certain statement cannot be proved from any ‘other’ proved statement and must be taken for granted to be true, then such a statement is called an axiom or a postulate.
Each branch of mathematics has its own populates or axioms. For example, the most fundamental axiom of geometry is that infinitely many lines can be drawn passing through a single point. The whole beautiful structure of geometry is based on five or six such axioms and every theorem in geometry can be ultimately Deducted from these axioms.
Argument, Premises and Conclusion
An argument is really speaking nothing more than an implication statement. Its hypothesis consists of the conjunction of several statements, called premises. In giving an argument, its premises are first listed (in any order), then connecting all, a conclusion is given. Example of an argument:
Premises: Every man is mortal.
Ram is a man.
Conclusion: Ram is mortal.
Symbolically, let us denote the premises of an argument by and its conclusion by . Then the argument is the statement . If this implication is true, the argument is valid otherwise it is invalid.
To be continued……
- Basic logic – connectives – NOT (gowers.wordpress.com)
- Welcome to the Cambridge Mathematical Tripos (gowers.wordpress.com)
[And if you're not serious about the test ...then read the whole post to know what the stupid author was trying to tell you. ]
STATEMENT: If the question you answered before you answered the question you answered after you answered the question you answered before you answered this one, was harder than the question you answered after you answered the question you answered before you answered this one, was the question you answered before you answered this one harder than this one? YES or NO? (more…)
Here is a list of 10 blogs which are my personal favorites at this time and I love reading them regularly:
- Richard Wiseman Blog
- Abstract Nonsense
- Mathematics and Multimedia
- Maths Is New Black
- Climbing Mount Bourbaki
- Waqas A Day
And Blog Of The Month for September is Richard Wiseman’s Blog located at http://richardwiseman.wordpress.com. He is an absolute puzzler, great author and his blog involves everything that one, who is not a math major, may also think and care about.
Psychologist and author Professor Richard Wiseman carries out research into luck, the science of self-help, perception, belief and deception.
His Latest Post is Five funny signs which he published just today.
Periodicity Of Content:10/10
Language: 10/10 [easiest]
Content Management: 9/10
Please note that this selection is personal and I have no affiliation with any organization. Your views are invited in form of comments. I have a huge list of other blogs at My Blogs Page. Have a look.
Last Updated: Sep. 6, 2011
14:24 India Standard Time
Four friends Matt, James, Ian and Barry, who all knew each other from being members of the Automattic, called Automatticians, sat around a table that had a dish with 11 apples in it. The chat was intense, and they ended up eating all the apples. Everybody had at least one apple, and everyone know that fact, and each automattician knew the number of apples he ate. They didn’t know how many apples each of the others ate, though. They agreed to ask only questions that they didn’t know the answers to:
Matt asked: Did you eat more apples than I did, James?
James: I don’t know. Did you, Ian, eat more apples than I did?
Ian: I don’t know.
Barry: Aha!! I figured out..
So, Barry figured out how many apples each person ate. Can you do the same?
Matt: 1 Apple
James: 2 Apples
Ian: 3 Apples
Barry: 5 Apples
Matt could not have eaten 5 or more. James could not have eaten only one or he would have known that he hadn’t eaten more than Ian. Neither could he have eaten 5 or more. He could have eaten 2 or 3 or 4 apples. Ian figures this out, although he still doesn’t know if he ate more than James. This mean that Ian must have eaten 3 or 4 apples. Barry can only deduce the other amounts if he ate 5 apples. And the rest, in order to add up to 11 , must have eaten 1, 2 and 3.
Inspired from a childhood heard puzzle.
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1. Simple Repitition Method
The classical method, very popular as in committing poems to memory by reading them over and over.
2. Full Concentration Method
Concentrate on the topic content while learning. Do not allow your mind to wander. Focus on names and numbers. Try deliberately to remember. Your approach should not be casual. Review soon after you learn, lest memory should fade away.
3. Visual Encoding Method
Translate information into visual formats like pictures, charts, diagrams, tables and graph.
4. Logical Organisation
Matter that is logically organised is retained better than the disjointed floating bits of information. Infuse meaning into what ever you learn. No “Nonsense Syllables”.
5. Mnemonics Method
Few good people try this method to learn some complicated series. There is no harm in using memory crutches like it, after grasping the spirit of the lesson. VIBGYOR is the most famous mnemonic that helps up to list the seven rainbow colors in their appropriate order. Once, I had remembered the name of the planets in the order of their distances from the sum by the acrostics “My very enlightened mother just served us noodles-in plates”, as it tells me the names of nine planets ( off course now they are eight, but I learned it earlier than the removal of pluto), in the order: Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune and Pluto.
6. Mock Teaching Method
If you find a particular portion of a chapter difficult to digest, imagine that you are teaching to a student sitting before you. Explain virtually the content of the lessen. The idea will get hammered into your mind.
7. Rhyming Method
Scholars say that verses are better than prose. Change any definition into verse and try to rhym them. Remember, I discuss just verses, and not poetry.
8. Chunking Method
Very popular method. You can divide a big list into a number of bits or chunks. For example, to remember a mobile number of ten digits, people usually split it to three or more chunks.
This involves the principal of association. First make a list of ten or twenty convenient pegs or key words that you can easily recall in the right sequence. For example, in alphabetical order, ant, butterfly, cat, dog, elephant, fox, and so on (Living beings are arranged). Cartoons also help in pegging.
This is the best method for that student, who want to learn very complicated topics. When you blog about any topic, you use all the nine methods listed above in your post. I got blogging, superior method to other nine.
- What is an easy way to remember the planets (wiki.answers.com)
One day a man, who looked like a tourist, came to a bicycle shop and bought a bicycle from a shop for $70. The cost price of the bicycle was $60. So the shopkeeper was happy that he had made a profit of $10 on the sale. However, at the time of setting the bill, the tourist offered to pay in travellers cheques as he had no cash money with him. The shopkeeper hesitated. He had no arrangement with the banks to encash travellers cheques. But he remembered that his friend, the shopkeeper next door, has such a provision, and so he took the cheques to his friend next door and got cash from him.
The travellers cheques were all of $20 each and so he had taken four cheques from the tourist totalling to $80. On encashing them the shopkeeper paid back the tourist the balance of $10.
The tourist happily climbed the bicycle and pedaled away whistling a tune.
However, the next morning shopkeeper’s friend who had taken the travellers cheques to the bank called on him and returned the cheques which had proved valueless and demanded the refund of his money. The shopkeeper quietly refunded the money to his friend and tried to trace the tourist who had given him the worthless cheques and taken away his bicycle. But the tourist could not be found.
How much did the shopkeeper lose altogether in this unfortunate transaction?
One can think of different answers for this question, but yet the correct answer is very simple. All we have to consider is that the shop owner could not have possibly lost more than the tourist actually stole.
The tourist got away with the bicycle which cost the shop owner $60 and the $10 ‘change’ , and therefore, he made off with $70. And this is the exact amount of the shopkeeper’s loss.
Source of The Puzzle:
This puzzle is l’ll modified than the original.