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Mathematical Reasoning

Blazing goIITian

Posted on
25 Apr 2009 11:31:18 IST
Posts: 1659
25 Apr 2009 11:31:18 IST
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Mathematical Reasoning
Engineering Entrance , JEE Main , JEE Advanced , Mathematics , academic , Mathematical Logic

Of  late many of us have been asking for books or sources to read theory regarding mathematical reasoning. Many of you may have googled it but didn't pay heed to this link :http://www.regentsprep.org/regents/math/math-a.cfm

So compiling it from there . Read it here else the link has been provided .

Types of Sentences

 One of the goals of studying mathematics is to develop the ability to think critically.  The study of critical thinking, or reasoning, is called logic.

All reasoning is based on the ways we put sentences together.  Let's start our examination of logic by defining what types of sentences we will be using.

mathematical sentence is one in which a fact or complete idea is expressed.  Because a mathematical sentence states a fact, many of them can be judged to be true or false.  Questions and phrases are not mathematical sentences since they cannot be judged to be true or false.

 "An isosceles triangle has two congruent sides." is a true mathematical sentence. "10 + 4 = 15" is a false mathematical sentence. "Did you get that one right?" is NOT a mathematical sentence - it is a question. "All triangles" is NOT a mathematical sentence - it is a phrase.
 There are two types of mathematical sentences: An open sentence is a sentence which contains a variable. "x + 2 = 8" is an open sentence -- the variable is "x." "It is my favorite color." is an open sentence-- the variable is "It." A closed sentence, or statement, is a mathematical sentence which can be judged to be true or false.  A closed sentence, or statement, has no variables. "Garfield is a cartoon character." is a true closed sentence, or statement. "A pentagon has exactly 4 sides." is a false closed sentence, or statement.

compound sentence is formed when two or more thoughts are connected in one sentence.

• "Today is a vacation day and I sleep late."

• "You can call me at 10 o'clock or you can call me at 2 o'clock."

• "If you are going to the beach, then you should take your sunscreen."

Negation(Not)

In logic, a negation of a statement is formed by placing the
word "not" into the original statement.  The negation will always have the opposite truth value of the original statement.
Under negation, what was TRUE, will become FALSE -

or - what was FALSE, will become TRUE.

Examples:

 1 Original Statement:  "15 + 20 equals 35."  (is true)Negation:  "15 + 20 does not equal 35."  (is false) 2 "A dog is a cat."  is a false statement."A dog is not a cat." is a true statement."It is not true that a dog is not a cat." is a false statement."It is not the case that it is not true that a dog is not a cat." is a true statement.*
 *While we do not usually talk in this manner because it is too confusing, we must be alert to people who attempt to win arguments by using several  negations at the same time.
 3. "A fish has gills." is a true statement."A fish does not have gills." is a false statement."It is not true that a fish does not have gills."  is a true statement.** ** Notice how using TWO negations, returns the truth value of the statement to its original value.  In plain English, this means that two negations will "undo" one another -- like a double negative.
 4 Original statement:  "Jedi masters do not use light sabers."Negation:  "Jedi masters do not not use light sabers." Better Negation:  "Jedi masters do use light sabers." Notice:  even though the first negation shows the proper insertion of the word "not", the second negation can be more easily read and understood.

Enrichment only:  (not tested on Math A or Math B)

Mathematicians often use symbols and tables to represent concepts in logic.  The use of these variables, symbols and tables creates a shorthand method for discussing logical sentences.

Truth table for negation (not):
(notice the symbol used for "not" in the table below)

 T F F T

truth table is a pictorial representation of all of the possible outcomes of the truth value of a sentence.  A letter such as  is used to represent the fact (or sentence).

 REMEMBER: Under negation, TRUE becomes FALSE - or - FALSE becomes TRUE.

Conjuction( And ,symbol - ^)

 In logic, a conjunction is a compound sentence formed by combining two sentences (or facts) using the word "and." A conjunction is true only when BOTH sentences(or facts) are true.

Examples:

1.  "Blue is a color and 7 + 3 = 10."  (and T = T)
Since both facts are true, the entire sentence is true.

2.  "One hour = exactly 55 minutes and one minute = exactly 60 seconds.
(F and T = F)  Since the first fact is false, the entire sentence is false.

3.  "3 + 4 = 6 and all dogs meow."  (F and F = F)
Since both facts are false, the entire sentence is false.

Enrichment only: (not tested on Math A or Math B)

Mathematicians often use symbols and tables to represent concepts in logic.  The use of these variables, symbols and tables creates a shorthand method for discussing logical sentences.

Truth table for conjunction (and):
(notice the symbol used for "and" in the table below)

 T T T T F F F T F F F F

truth table is a pictorial representation of all of the possible outcomes of the truth value of a compound sentence.  Letters such as and  are used to represent the facts (or sentences) within the compound sentence.

 Remember:   For a conjunction (and) to be true, BOTH factsmust be true.

Disjunction (OR,symbol - V)

 In logic, a disjunction is a compound sentence formed by combining two sentences (or facts) using the word "or." A disjunction is true when EITHER or BOTH sentences (or facts) are true.

Examples:

 1 "Blue is a color or 7 + 3 = 10."(T or T = T)  Since both facts are true, the entire sentence is true. 2 "One hour = exactly 55 minutes or one minute = exactly 60 seconds."(F or T = T)  Since the second fact is true, the entire sentence is true. 3 "3 + 4 = 6 or all dogs meow."(F or F = F) Since both facts are false, the entire sentence is false. 4 "The word cat has 3 letters or the word dog has four letters." (T  or F = T)  Since the first fact is true, the entire sentence is true.

Enrichment only: (not tested on Math A or Math B)

Mathematicians often use symbols and tables to represent concepts in logic.  The use of these variables, symbols and tables creates a shorthand method for discussing logical sentences.

Truth table for disjunction (or):
(notice the symbol used for "or" in the table below)

 T T T T F T F T T F F F

truth table is a pictorial representation of all of the possible outcomes of the truth value of a compound sentence.  Letters such as and  are used to represent the facts (or sentences) within the compound sentence.

Remember:

 For a disjunction (or) to be true, EITHER or BOTH facts must be true.

BiConditional - if and only if

 In logic, a biconditional is a compound statement formed by combining two conditionals under "and."  Biconditionals are true when both statements (facts) have the exact same truth value.

A biconditional is read as "[some fact]  if and only if [another fact]" and is true when the truth values of both facts are exactly the same
-- BOTH TRUE or BOTH FALSE.

 Biconditionals are oftenused to form definitions.

Definition:  A triangle is isosceles if and only if the triangle has two congruent (equal) sides.

The "if and only if" portion of the definition tells you that the statement is true when either sentence (or fact) is the hypothesis.  This means that both of the statements below are true:

If a triangle is isosceles, then the triangle has two
congruent (equal) sides.
(true)

If a triangle has two congruent (equal) sides,
then the triangle is isosceles.
(true)

Enrichment only: (not tested on Math A or Math B)

Mathematicians often use symbols and tables to represent concepts in logic.  The use of these variables, symbols and tables creates a shorthand method for discussing logical sentences.

Truth table biconditional (if and only if):
(notice the symbol used for "if and only if" in the table below)

 T T T T F F F T F F F T

truth table is a pictorial representation of all of the possible outcomes of the truth value of a compound sentence.  Letters such as and  are used to represent the facts (or sentences) within the compound sentence.

Conditional - if...then

 In logic, a conditional is a compound statement formed by combining two sentences (or facts) using the words "if ... then."   A conditional can also be called an implication.

The truth values for a conditional (implication) are hard to remember.
You will want to study this section.

 The example below may help you remember the truth values for the conditional:

The statement:

Your teacher tells you that
"if you participate in class, then you will get extra points."

 fact 1:   "you participate in class."fact 2:   "you get participation points."

When is the teacher's statement true?

1.  If you participate in class (fact 1 true) and you get extra points (fact 2 true
then the teacher's statement is true.

2.  If you participate in class (fact 1 true) and you do not get extra points
(fact 2 false), then the teacher did not tell the truth and the statement is false.

3.  If you do not participate in class (fact 1 false), we cannot judge the truth
of the teacher's statement.  The teacher did not tell you what would happen
if you did NOT participate in class.  Since we cannot accuse the teacher of
making a false statement, we assign "true" to the statement.

"If you participate in class, then you will get extra points."
will be true in all cases except one:  when you participate in class and you do NOT get the extra points.

Enrichment only: (not tested on Math A or Math B)

Mathematicians often use symbols and tables to represent concepts in logic.  The use of these variables, symbols and tables creates a shorthand method for discussing logical sentences.

Truth table for conditional (if...then):
(notice the symbol used for "if...then" in the table below)

 T T T T F F F T T F F T

truth table is a pictorial representation of all of the possible outcomes of the truth value of a compound sentence.  Letters such as and  are used to represent the facts (or sentences) within the compound sentence.

 REMEMBER: IF...THEN is only FALSEwhen T implies F. All other cases are TRUE.

Converse

 The converse of a conditional statement is formed byinterchanging the hypothesis and conclusion of the original statement.In other words, the parts of the sentence change places.The words "if" and "then" do not move.

Example:

Conditional:  "If the space shuttle was launchedthen a cloud of smoke was seen."

Converse:  "If a cloud of smoke was seenthen the space shuttle was launched."

 HINT:  Try to associate the logical CONVERSE withConverse™ sneakers -- think of the two parts of the sentence "putting on their sneakers" and "running" to their new positions.

 ** It is important to remember that the converse does NOT necessarily have the same truth value as the original conditional statement.

Consider:

Conditional:  "If the space shuttle was launchedthen a cloud of smoke was seen."  This statement is true since the exhaust from the shuttle creates a cloud of smoke.

Converse:  "If a cloud of smoke was seenthen the space shuttle was launched."  This statement is not always true since many other events (a fire, a running herd of buffalo, car exhaust, etc.) could have caused a cloud of smoke.

 An interesting fact:  The converse has the same truth value as the inverse of the original statement.  The CONVERSE and the INVERSE of the original statement are logically equivalent.("equivalent" means "the same")

Enrichment only: (not tested on Math A or Math B)

A truth table clearly shows the relationship between the conditional, the converse, and the inverse:

 Conditional Converse Inverse T T F F T T T T F F T F T T F T T F T F F F F T T T T T

Inverse :

 The inverse of a conditional statement is formed bynegating the hypothesis and negating the conclusion of the original statement.In other words, the word "not" is added to both parts of the sentence.

Example:

Conditional:  "If you grew up in Alaskathen you have seen snow."

Inverse:  "If you did not grow up in Alaskathen you have not seen snow."

 HINT:  Remember that to create an INverse, you will needto INsert the word NOT into both portions of the sentence.Since you are actually negating each part of the sentence,you may also use other words (in addition to NOT)to create the negation.
 ** It is important to remember that the inverse does NOT necessarily have the same truth value as the original conditional statement.

Consider:

Conditional:  "If you grew up in Alaskathen you have seen snow."
Considering the climatic conditions in Alaska, this statement is true.

Inverse:  "If you did not grow up in Alaskathen you have not seen snow."  Considering that there are other areas in the world that have snow (such as New York state), this statement is false.

 An interesting fact: The inverse has the same truth value as the converse of the original statement.  The  INVERSE and the CONVERSE of the original statement are logically equivalent.("equivalent" means "the same")

Enrichment only: (not tested on Math A or Math B)

A truth table clearly shows the relationship between the conditional, the converse, and the inverse:

 Conditional Converse Inverse T T F F T T T T F F T F T T F T T F T F F F F T T T T T

Contrapositive :

 The contrapositive of a conditional statement is formed bynegating both the hypothesis and the conclusion, and then interchangingthe resulting negations. In other words, the contrapositive negates and switches the parts of the sentence.  It does BOTH the jobs of the INVERSE and the CONVERSE.

Example:

Conditional:
"If 9 is an odd numberthen 9 is divisible by 2."
(true)                               (false)
 This statement is logically FALSE.

Contrapositive
"If 9 is not divisible by 2then 9 is not an odd number."
(true)                                 (false)
 This statement is logically FALSE.

 HINT:  Remember that the contrapositive (a big long word) is really the combining together of the strategies of two other words:  converse and inverse.

 An important fact to remember about the contrapositive, is that it always has the SAME truth value as the original conditional statement.

**If the original statement is TRUE, the contrapositive is TRUE.
If the original statement is FALSE, the contrapositive is FALSE.

They are said to be
logically equivalent.
("equivalent" means "the same")

Enrichment only: (not tested on Math A or Math B)

A truth table can be used to show that a conditional statement and its contrapositive are logically equivalent.  Notice that the truth values are the same.

 Conditional Contrapositive T T T T T F F F F T T T F F T T

 Remember:The contrapositive is the mixing of the inverse and the converse.

Compound Sentences:

compound sentence is formed when
two or more thoughts are connected in one sentence.

 The following are examples of compound sentences: "21 is  divisible by 3 and 21 is not prime." "45 is a multiple of 9 or 13 - 20 = 7." "If 4 + 6 = 10 and 3 + 3 = 9, then all rectangles are squares."

When attempting to determine the truth value of a compound sentence, first determine the truth value of each of the components of the sentence.
Let's examine the examples listed above.

1.  Determine the truth value of:    "21 is divisible by 3 and 21 is not prime."

 "21 is divisible by 3"  (true)"21 is not prime"  (true)
 Substitute the truth values for the facts: T and T Simplify the conjunction (and): T Answer:  The compound sentence (statement) is true.

2.  Determine the truth value of:    "45 is a multiple of 9 or 13 - 20 = 7."

"45 is a multiple of 9" (true)
"13 - 20 = 7" (false)

 Substitute the truth values for the facts: T or F Simplify the disjunction (or): T Answer:  The compound sentence (statement) is true.

3.  Determine the truth value of:
"If 4 + 6 = 10 and 3 + 3 = 9, then all rectangles are squares."

"4 + 6 = 10 " (true)
"3 + 3 = 9" (false)
"all rectangles are squares." (false)

 Substitute the truth values for the facts: if (T and F) then F Simplify the conjunction (AND) first: if F then F Simplify the conditional: T Answer:  The compound sentence (statement) is true.

Enrichment only: (not tested on Math A or Math B)

 When the truth value of one or more of the components of a compound sentence is unknown, all of the possible truth values must be considered.  A truth table is the easiest way to show all of these possibilities.

Construct a truth table for

 T T T T F F T F F T F T F T F T F T F F F F T T

The truth table tells you that the compound sentence will be false only when pand q are both true.  In all other situations, the compound sentence is true.

#### Comments (6)

Blazing goIITian

Joined: 6 Aug 2007 16:16:05 IST
Posts: 1860
25 Apr 2009 11:39:39 IST
0 people liked this

gud revision...

Hot goIITian

Joined: 21 Apr 2009 18:56:01 IST
Posts: 155
25 Apr 2009 11:42:12 IST
0 people liked this

excellent yaar

Blazing goIITian

Joined: 18 Feb 2009 20:57:46 IST
Posts: 902
25 Apr 2009 12:25:10 IST
0 people liked this

a nice article

New kid on the Block

Joined: 12 May 2012 14:13:41 IST
Posts: 1
12 May 2012 14:14:10 IST
0 people liked this

tautology is not there!!:(((

New kid on the Block

Joined: 12 May 2012 14:13:53 IST
Posts: 3
12 May 2012 14:16:58 IST
0 people liked this

tautology is not in this article!!!:((

Blazing goIITian

Joined: 8 May 2012 12:29:36 IST
Posts: 466
12 May 2012 14:35:46 IST
0 people liked this

gud

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