Thursday, June 22, 2023

Chapter 14 - Mathematical Reasoning

In the previous section, we completed a discussion on limits and derivatives. In this chapter, we will see mathematical reasoning.

Some basics about reasoning can be written in 4 steps:
1. Human beings have the ability to engage in the process of reasoning.
• The "process of reasoning" is the process in which we think (in a logical way) about some thing. We think about some thing to arrive at a conclusion.
2. Let us see some examples:
• Early human beings thought about the “cause of lightning”. Other animals are not able to engage in such a thought.
• Early human beings thought about “why the pebbles found in river beds are rounded ?”. Other animals are not able to engage in such a thought.
• Human beings at present are thinking about “what is inside a proton ?”.  Other animals are not able to engage in such a thought.
3. The process of thinking must be done in a logical and sensible way. Then only we can call it reasoning.
4. On seeing a light above a distant hill, a person may think for a while and come to the conclusion that, the light is caused by some super natural power.
• Such thinking process cannot be called reasoning.
• We must investigate and analyze all available facts. Also, we must try to obtain new facts.


• Reasoning can be done in mathematical problems also.
• We have see inductive reasoning in chapter 4. In this chapter we will see deductive reasoning.
• The basic unit required for mathematical reasoning is a statement.
 

• Some basics about “statement” in mathematics can be written in 6 steps
1. Consider the following three sentences:
(i) In 2021, the President of USA was a woman.
(ii) A blue whale weighs more than an elephant.
(iii) This building is beautiful.
2. Let us analyze each sentence:
(i) When we read the first sentence, we immediately decide that, the sentence is false.
No doubt arises in our mind. That is., we can write with out any doubt that, the sentence is false.     
(ii) When we read the second sentence, we immediately decide that, the sentence is true.
No doubt arises in our mind. That is., we can write with out any doubt that, the sentence is true.
(iii) When we read the third sentence, some doubt can arise.
    ♦ Some people may consider the building to be beautiful.
    ♦ Some other people may consider the building to be ordinary.
    ♦ Yet some others may consider the building to be ugly.
• For a person who consider the building to be beautiful, the sentence is true.
• For a person who consider the building to be ordinary or ugly, the sentence is false.
• In such cases,
    ♦ We cannot say that the sentence is true.
    ♦ We cannot say that the sentence is false.
• That means:
There is a possibility that, the sentence is both true and false.
• In such situations, we say that:
    ♦ the sentence is ambiguous.
    ♦ or we say that: there is ambiguity in the sentence.
(The words ambiguous or ambiguity refers to the situation where a sentence has more than one possible meaning)
3. Now we can write a basic idea about statements:
• If there is no ambiguity in a sentence, then that sentence can be considered as a statement in mathematics.
• If there is ambiguity in a sentence, then that sentence is not considered as a statement in mathematics.
4. Let us consider the three sentences that we wrote in step (1).
• Sentences (i) and (ii) can be considered as statements in mathematics.
• Sentence (iii) is not considered as a statement in mathematics.
5. Let us write a precise definition for statement. It can be written in 3 steps:
(i) A sentence is called a mathematically acceptable statement if it is true.
(ii) A sentence is called a mathematically acceptable statement if it is false.
(iii) A sentence is not called a mathematically acceptable statement if it is both true and false.
6. From now on wards, whenever we say “statement”, it should be understood that, it is a mathematically acceptable statement.


Let us see a few more examples:
1. Consider the sentence:
Three plus five equal eight.
• This sentence is true. There is no ambiguity. So it is a statement.
2. Consider the sentence:
The sum of two positive numbers is positive.
• This sentence is true. There is no ambiguity. So it is a statement.
3. Consider the sentence:
All prime numbers are odd numbers.
• This sentence is false. There is no ambiguity. So it is a statement.  
(Recall that there is one and only one even prime number, which is ‘2’)
4. Consider the sentence:
The sum of x and y is greater than zero.
• On some occasions, this sentence is true.
   ♦ For example, when x = 3 and y = 8
• On some other occasions, this sentence is false.
   ♦ For example, when x = 3 and y = -8
• There is ambiguity in this sentence. So it is not a statement.
5. Consider the sentence:
For any two natural numbers x and y, the sum (x+y) is greater than zero.  
• This sentence is true. There is no ambiguity. So it is a statement.  
(Recall that natural numbers are: 1, 2, 3, 4, . . .)
6. Consider the sentence:
How beautiful !
• This is an exclamatory sentence. Exclamatory sentences convey strong emotion or excitement. They are not considered as statements in mathematics.
7. Consider the sentence:
Open the door.
• This sentence is an order. Orders are not considered as statements in mathematics.
8. Consider the sentence:
Where are you going ?
• This sentence is a question. Questions are not considered as statements in mathematics.
9. Consider the sentence:
Tomorrow is Friday.
• On some occasions (Thursdays), this sentence is true.
• On some other occasions (days other than Thursdays), this sentence is false.
• There is ambiguity in this sentence. So it is not a statement.
◼ Word “tomorrow” indicates variable time. It can be any day of the week. It depends on the day on which the sentence is written. Similar is the case with the words “today”,  and “yesterday”. If any of these three words are present in a sentence, then it cannot be considered as a statement in mathematics.
10. Consider the sentence:
Kashmir is far from here.
• On some occasions, this sentence is true.
For example, when the sentence is written at a place far away from Kashmir.
• On some other occasions, this sentence is false.
For example, when the sentence is written at a place close to Kashmir.
• There is ambiguity in this sentence. So it is not a statement.
◼ Word “here” indicates variable position. It can be any position in space. It depends on the position at which the sentence is written. Similar is the case with the word “there”. If any of these two words are present in a sentence, then it cannot be considered as a statement in mathematics.
11. Consider the sentence:
She is a mathematics graduate.
• On some occasions, this sentence is true.
For example, when the person represented by the pronoun “she” is a mathematics graduate.
• On some other occasions, this sentence is false.
For example, when the person represented by the pronoun “she” is not a mathematics graduate .
• There is ambiguity in this sentence. So it is not a statement.
◼ Pronoun “she” indicates variable person. It can be any person. Similar is the case with the pronouns “he”, “it”, “they” etc., If pronouns are present in a sentence, then it cannot be considered as a statement in mathematics.
12. Consider the sentence:
There are 30 days in a month.
• On some occasions, this sentence is true.
For example, when the month being referred to, is April.
• On some occasions, this sentence is false.
For example, when the month being referred to, is August.
• There is ambiguity in this sentence. So it is not a statement.
13. Consider the sentence:
There are 40 days in a month.
• This sentence is always false. There is no ambiguity. So it is a statement.
14.  Consider the sentence:
(x+y)2 = x2 + y2.
• On some occasions, this sentence is true.
   ♦ For example, when x = 1 and y = 0
• On some other occasions, this sentence is false.
   ♦ For example, when x = 2 and y = 3
• There is ambiguity in this sentence. So it is not a statement.
15. Consider the sentence:
There exists two real numbers x and y, such that (x+y)2 = x2 + y2.  
• This sentence is true. There is no ambiguity. So it is a statement.


• Statements are denoted by small letters p, q, r etc.,
• For example, consider the statement:
For any two natural numbers x and y, the sum (x+y) is greater than zero.  
• We can write:
p: For any two natural numbers x and y, the sum (x+y) is greater than zero.


Let us see some solved examples:

Solved example 14.1
Check whether the following sentences are statements. Give reasons for your answer.
(i) 8 is less than 6.
(ii) Every set is a finite set.
(iii) The sun is a star.
(iv) Mathematics is fun.
(v) There is no rain without clouds.
(vi) How far is Chennai from here ? 
Solution:
Part (i):
8 is less than 6.
• This statement is false. There is no ambiguity. So it is a statement.
Part (ii):
Every set is a finite set.
• This statement is false. There is no ambiguity. So it is a statement.
(There are sets which are not finite)
Part (iii):
The sun is a star.
• This statement is true. There is no ambiguity. So it is a statement.
(It is scientifically proven that, sun is a star)
Part (iv):
Mathematics is fun.
• For a person who is interested in mathematics, the sentence is true.
• For a person who is not interested in mathematics, the sentence is false.
• There is ambiguity in this sentence. So it is not a statement.
Part (v):
There is no rain without clouds.
• This statement is true. There is no ambiguity. So it is a statement.
(It is scientifically proven that, clouds are required for rain to occur)
Part (vi):
How far is Chennai from here ?
• This is a question. Also, the word "here" is present. So it is not a statement.


From the above discussion and examples, the following three points can be noted:
1. We will be asked to say whether a given sentence is a statement or not.
2. The answer can be any one of the two given below:
Yes. The given sentence is a statement.
No. The given sentence is not a statement.
3. But besides answering “Yes” or “No”, we must write the reason also.


The link to a few more solved examples is given below:

Exercise 14.1


In the next section, we will see New statements from old.

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