vThere are few things which we know which are not capable of
mathematical reasoning and when these can not, it is a sign that our
knowledge of them is very small and confused and where a mathematical
reasoning can be had, it is as great a folly to make use of another,
as to grope for a thing in the dark when you have a candle stick
standing by you. – ARTHENBOT v
14.1 Introduction
In this Chapter, we shall discuss about some basic ideas of
Mathematical Reasoning. All of us know that human beings
evolved from the lower species over many millennia. The
main asset that made humans “superior” to other species
was the ability to reason. How well this ability can be used
depends on each person’s power of reasoning. How to
develop this power? Here, we shall discuss the process of
reasoning especially in the context of mathematics.
In mathematical language, there are two kinds of
reasoning – inductive and deductive. We have already
discussed the inductive reasoning in the context of
mathematical induction. In this Chapter, we shall discuss
some fundamentals of deductive reasoning.
14.2 Statements
The basic unit involved in mathematical reasoning is a mathematical statement.
Let us start with two sentences:
In 2003, the president of India was a woman.
An elephant weighs more than a human being.
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George Boole
(1815 - 1864)
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When we read these sentences, we immediately decide that the first sentence is
false and the second is correct. There is no confusion regarding these. In mathematics
such sentences are called statements.
On the other hand, consider the sentence:
Women are more intelligent than men.
Some people may think it is true while others may disagree. Regarding this sentence
we cannot say whether it is always true or false . That means this sentence is ambiguous.
Such a sentence is not acceptable as a statement in mathematics.
A sentence is called a mathematically acceptable statement if it is either
true or false but not both. Whenever we mention a statement here, it is a
mathematically acceptable” statement.
While studying mathematics, we come across many such sentences. Some examples
are:
Two plus two equals four.
The sum of two positive numbers is positive.
All prime numbers are odd numbers.
Of these sentences, the first two are true and the third one is false. There is no
ambiguity regarding these sentences. Therefore, they are statements.
Can you think of an example of a sentence which is vague or ambiguous? Consider
the sentence:
The sum of x and y is greater than 0
Here, we are not in a position to determine whether it is true or false, unless we
know what
x and y are. For example, it is false where x = 1, y = –3 and true when
x = 1 and y = 0. Therefore, this sentence is not a statement. But the sentence:
For any natural numbers x and y, the sum of x and y is greater than 0
is a statement.
Now, consider the following sentences :
How beautiful!
Open the door.
Where are you going?
Are they statements? No, because the first one is an exclamation, the second
an order and the third a question. None of these is considered as a statement in
mathematical language. Sentences involving variable time such as “today”, “tomorrow”
or “yesterday” are not statements. This is because it is not known what time is referred
here. For example, the sentence
Tomorrow is Friday
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is not a statement. The sentence is correct (true) on a Thursday but not on other
days. The same argument holds for sentences with pronouns unless a particular
person is referred to and for variable places such as “here”, “there” etc., For
example, the sentenc
es
She is a mathematics graduate.
Kashmir is far from here.
are not statements.
Here is another sentence
There are 40 days in a month.
Would you call this a statement? Note that the period mentioned in the sentence
above is a “variable time” that is any of 12 months. But we know that the sentence is
always false (irrespective of the month) since the maximum number of days in a month
can never exceed 31. Therefore, this sentence is a statement. So, what makes a sentence
a statement is the fact that the sentence is either true or false but not both.
While dealing with statements, we usually denote them by small letters p, q, r
,...
For example, we denote the statement “Fire is always hot” by
p. This is also written
as
p: Fire is always hot.
Example 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
(i) This sentence is false because 8 is greater than 6. Hence it is a statement.
(ii) This sentence is also false since there are sets which are not finite. Hence it is
a statement.
(iii) It is a scientifically established fact that sun is a star and, therefore, this sentence
is always true. Hence it is a statement.
(iv) This sentence is subjective in the sense that for those who like mathematics, it
may be fun but for others it may not be. This means that this sentence is not always
true. Hence it is not a statement.
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(v) It is a scientifically established natural phenomenon that cloud is formed before it
rains. Therefore, this sentence is always true. Hence it is a statement.
(vi) This is a question which also contains the word “Here”. Hence it is not a statement.
The above examples show that whenever we say that a sentence is a statement
we should always say why it is so. This “why” of it is more important than the answer.
EXERCISE 14.1
1. Which of the following sentences are statements? Give reasons for your answer.
(i) There are 35 days in a month.
(ii) Mathematics is difficult.
(iii) The sum of 5 and 7 is greater than 10.
(iv) The square of a number is an even number.
(v) The sides of a quadrilateral have equal length.
(vi) Answer this question.
(vii) The product of (–1) and 8 is 8.
(viii) The sum of all interior angles of a triangle is 180°.
(ix) Today is a windy day.
(x) All real numbers are complex numbers.
2. Give three examples of sentences which are not statements. Give reasons for the
answers.
14.3 New Statements from Old
We now look into method for producing new statements from those that we already
have. An English mathematician, “George Boole” discussed these methods in his book
“The laws of Thought” in 1854. Here, we shall discuss two techniques.
As a first step in our study of statements, we look at an important technique that
we may use in order to deepen our understanding of mathematical statements. This
technique is to ask not only what it means to say that a given statement is true but also
what it would mean to say that the given statement is not true.
14.3.1 Negation of a statement The denial of a statement is called the negation of
the statement.
Let us consider the statement:
p: New Delhi is a city
The negation of this statement is
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It is not the case that New Delhi is a city
This can also be written as
It is false that New Delhi is a city.
This can simply be expressed as
New Delhi is not a city.
Definition
1 If p is a statement, then the negation of p is also a statement and is
denoted by p, and read as ‘not p’.
A
Note While forming the negation of a statement, phrases like, “It is not the
case” or “It is false that” are also used.
Here is an example to illustrate how, by looking at the negation of a statement, we
may improve our understanding of it.
Let us consider the statement
p: Everyone in Germany speaks German.
The denial of this sentence tells us that not everyone in Germany speaks German.
This does not mean that no person in Germany speaks German. It says merely that at
least one person in Germany does not speak German.
We shall consider more examples.
Example 2 Write the negation of the following statements.
(i) Both the diagonals of a rectangle have the same length.
(ii)
7
is rational.
Solution (i) This statement says that in a rectangle, both the diagonals have the same
length. This means that if you take any rectangle, then both the diagonals have the
same length. The negation of this statement is
It is false that both the diagonals in a rectangle have the same length
This means the statement
There is atleast one rectangle whose both diagonals do not
have the same length.
(ii) The negation of the statement in (ii) may also be written as
It is not the case that
7
is rational.
This can also be rewritten as
7
is not rational.
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Example 3 Write the negation of the following statements and check whether the
resulting statements are true,
(i) Australia is a continent.
(ii) There does not exist a quadrilateral which has all its sides equal.
(iii) Every natural number is greater than 0.
(iv) The sum of 3 and 4 is 9.
Solution
(i) The negation of the statement is
It is false that Australia is a continent.
This can also be rewritten as
Australia is not a continent.
We know that this statement is false.
(ii) The negation of the statement is
It is not the case that there does not exist a quadrilateral which has all its sides
equal.
This also means the following:
There exists a quadrilateral which has all its sides equal.
This statement is true because we know that square is a quadrilateral such that its four
sides are equal.
(iii) The negation of the statement is
It is false that every natural number is greater than 0.
This can be rewritten as
There exists a natural number which is not greater than 0.
This is a false statement.
(iv) The negation is
It is false that the sum of 3 and 4 is 9.
This can be written as
The sum of 3 and 4 is not equal to 9.
This statement is true.
14.3.2 Compound statements Many mathematical statements are obtained by
combining one or more statements using some connecting words like “and”, “or”, etc.
Consider the following statement
p: There is something wrong with the bulb or with the wiring.
This statement tells us that there is something wrong with the bulb or there is
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something wrong with the wiring. That means the given statement is actually made up
of two smaller statements:
q: There is something wrong with the bulb.
r: There is something wrong with the wiring.
connected by “or”
Now, suppose two statements are given as below:
p: 7 is an odd number.
q: 7 is a prime number.
These two statements can be combined with “and”
r: 7 is both odd and prime number.
This is a compound statement.
This leads us to the following definition:
Definition 2
A Compound Statement
is a statement which is made up of two or
more statements. In this case, each statement is called a component statement.
Let us consider some examples.
Example 4
Find the component statements of the following compound statements.
(i) The sky is blue and the grass is green.
(ii) It is raining and it is cold.
(iii) All rational numbers are real and all real numbers are complex.
(iv) 0 is a positive number or a negative number.
Solution
Let us consider one by one
(i) The component statements are
p: The sky is blue.
q: The grass is green.
The connecting word is ‘and’.
(ii) The component statements are
p: It is raining.
q: It is cold.
The connecting word is ‘and’.
(iii) The component statements are
p: All rational numbers are real.
q: All real numbers are complex.
The connecting word is ‘and’.
(iv)The component statements are
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p: 0 is a positive number.
q: 0 is a negative number.
The connecting word is ‘or’.
Example 5 Find the component statements of the following and check whether they
are true or not.
(i) A square is a quadrilateral and its four sides equal.
(ii) All prime numbers are either even or odd.
(iii) A person who has taken Mathematics or Computer Science can go for
MCA.
(iv) Chandigarh is the capital of Haryana and UP.
(v)
2
is a rational number or an irrational number.
(vi) 24 is a multiple of 2, 4 and 8.
Solution (i) The component statements are
p: A square is a quadrilateral.
q: A square has all its sides equal.
We know that both these statements are true. Here the connecting word is ‘and’.
(ii) The component statements are
p: All prime numbers are odd numbers.
q: All prime numbers are even numbers.
Both these statements are false and the connecting word is ‘or’.
(iii) The component statements are
p: A person who has taken Mathematics can go for MCA.
q: A person who has taken computer science can go for MCA.
Both these statements are true. Here the connecting word is ‘or’.
(iv) The component statements are
p: Chandigarh is the capital of Haryana.
q: Chandigarh is the capital of UP.
The first statement is true but the second is false. Here the connecting word is ‘and’.
(v) The component statements are
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p:
2
is a rational number.
q:
2
is an irrational number.
The first statement is false and second is true. Here the connecting word is ‘or’.
(vi) The component statements are
p: 24 is a multiple of 2.
q: 24 is a multiple of 4.
r: 24 is a multiple of 8.
All the three statements are true. Here the connecting words are ‘and’.
Thus, we observe that compound statements are actually made-up of two or more
statements connected by the words like “and”, “or”, etc. These words have special
meaning in mathematics. We shall discuss this mattter in the following section.
EXERCISE 14.2
1. Write the negation of the following statements:
(i) Chennai is the capital of Tamil Nadu.
(ii)
2
is not a complex number
(iii) All triangles are not equilateral triangle.
(iv) The number 2 is greater than 7.
(v) Every natural number is an integer.
2. Are the following pairs of statements negations of each other:
(i) The number x is not a rational number.
The number x is not an irrational number.
(ii) The number x is a rational number.
The number x is an irrational number.
3. Find the component statements of the following compound statements and check
whether they are true or false.
(i) Number 3 is prime or it is odd.
(ii) All integers are positive or negative.
(iii) 100 is divisible by 3, 11 and 5.
14.4 Special Words/Phrases
Some of the connecting words which are found in compound statements like “And”,
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“Or”, etc. are often used in Mathematical Statements. These are called connectives.
When we use these compound statements, it is necessary to understand the role of
these words. We discuss this below.
14.4.1 The word “And”
Let us look at a compound statement with “And”.
p: A point occupies a position and its location can be determined.
The statement can be broken into two component statements as
q: A point occupies a position.
r: Its location can be determined.
Here, we observe that both statements are true.
Let us look at another statement.
p: 42
is divisible by 5, 6 and 7.
This statement has following component statements
q: 42 is divisible by 5.
r: 42 is divisible by 6.
s: 42 is divisible by 7.
Here, we know that the first is false while the other two are true.
We have the following rules regarding the connective “And”
1. The compound statement with ‘And’ is true if all its component
statements are true.
2. The component statement with ‘And’ is false if any of its component
statements is false (this includes the case that some of its component
statements are false or all of its component statements are false).
Example 6
Write the component statements of the following compound statements
and check whether the compound statement is true or false.
(i) A line is straight and extends indefinitely in both directions.
(ii) 0 is less than every positive integer and every negative integer.
(iii) All living things have two legs and two eyes.
Solution (i) The component statements are
p: A line is straight.
q: A line extends indefinitely in both directions.
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Both these statements are true, therefore, the compound statement is true.
(ii) The component statements are
p: 0 is less than every positive integer.
q: 0 is less than every negative integer.
The second statement is false. Therefore, the compound statement is false.
(iii) The two component statements are
p: All living things have two legs.
q: All living things have two eyes.
Both these statements are false. Therefore, the compound statement is false.
Now, consider the following statement.
p: A mixture of alcohol and water can be separated by chemical methods.
This sentence cannot be considered as a compound statement with “And”. Here the
word “And” refers to two things – alcohol and water.
This leads us to an important note.
A
Note Do not think that a statement with “And” is always a compound statement
as shown in the above example. Therefore, the word “And” is not used as a connective.
14.4.2 The word “Or”
Let us look at the following statement.
p: Two lines in a plane either intersect at one point or they are parallel.
We know that this is a true statement. What does this mean? This means that if two
lines in a plane intersect, then they are not parallel. Alternatively, if the two lines are not
parallel, then they intersect at a point. That is this statement is true in both the situations.
In order to understand statements with “Or” we first notice that the word “Or” is
used in two ways in English language. Let us first look at the following statement.
p: An ice cream or pepsi is available with a Thali in a restaurant.
This means that a person who does not want ice cream can have a pepsi along
with
Thali or one does not want pepsi can have an ice cream along with Thali. That is,
who do not want a pepsi can have an ice cream. A person cannot have both ice cream
and pepsi. This is called an exclusive “Or”.
Here is another statement.
A student who has taken biology or chemistry can apply for M.Sc.
microbiology programme.
Here we mean that the students who have taken both biology and chemistry can
apply for the microbiology programme, as well as the students who have taken only
one of these subjects. In this case, we are using inclusive “Or”.
It is important to note the difference between these two ways because we require this
when we check whether the statement is true or not.
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