Saturday, June 24, 2023

Chapter 14.1 - Negation of A Statement

In the previous section, we saw what a "statement" is. In this section, we will see new statements from old.

Some basic points can be written in 3 steps:
1. We are discussing about mathematical reasoning.
• We saw “statements” which are the basic units of mathematical reasoning.
• We want to know how statements help us in the process of mathematical reasoning.
2. The English mathematician George Boole put forward an effective technique to analyze statements.
• In this technique, we ask two questions:
   ♦ What happens when the statement is true ?
   ♦ What happens when the statement is false ?
3. Finding answers to the above two questions is a major step in mathematical reasoning.


So our next aim is to learn about Negation of a statement. This can be written in 3 steps:
1. The denial of a statement is called the negation of the statement.
2. Let us see an example. It can be written in 3 steps:
(i) Consider the following statement:
p: New Delhi is a city.
(ii) The negation of this statement can be written in any one of the three ways shown below:
• It is not the case that New Delhi is a city
• It is false that New Delhi is a city
• New Delhi is not a city.
(iii) If p is a statement, then negation of p is also a statement.
   ♦ It is denoted as ~p.
   ♦ It is read as ‘not p’.
3. ~p helps us to analyze p in a better way. This can be demonstrated using some examples.
Example 1:
This can be written in 4 steps:
(i) Consider the statement:
p: Every one in Germany speaks German.   
(ii) We want ~p.
• p tells us that all people living in Germany speak German.
• ~p must tell us that all people living in Germany does not speak German.
(iii) This ~p can be achieved by saying that there is at least one person in Germany, who does not speak that language.
• So we get:
~p: There is at least one person in Germany who does not speak German.
(iv) p and ~p enable us to think of various possibilities.

Example 2:
This can be written in 4 steps:
(i) Consider the statement:
p: Both the diagonals of a rectangle have the same length.   
(ii) We want ~p.
• p tells us that: Take any rectangle. Both it's diagonals will be of the same length.
• ~p must tell us that the statement is not true for any rectangle.
(iii) This ~p can be achieved by saying that there is at least one rectangle where both diagonals are not the same.
• So we get:
~p: There is at least one rectangle where both diagonals are not of the same length.
(iv) p and ~p enable us to think of various possibilities.

Example 3:
This can be written in 4 steps:
(i) Consider the statement:
p: √7 is rational.   
(ii) We want ~p.
• p tells us that: √7 is rational.
• ~p must tell us that the statement is not true.
(iii) This ~p can be achieved by saying that it is not rational.
• So we get:
~p: √7 is not rational.
(iv) p and ~p enable us to think of various possibilities.


Now we will see some solved examples.

Solved example 14.2
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 it’s sides equal.
(iii) The sum of 3 and 4 is 9.
Solution:
Part (i):
1. Consider the statement:
p: Australia is a continent.
2. The negation can be written as:
~p: It is false that Australia is a continent.
OR
~p: Australia is not a continent.
3. ~p is false.

Part (ii):
1. Consider the statement:
p: There does not exist a quadrilateral which has all it’s sides equal.
2. We want ~p.
• p tells us that: Not even a single quadrilateral (with the given property) exists.
• ~p must tell us that "not even a single" is false.
3. This ~p can be achieved by saying that there is at least one quadrilateral.
• So we get:
~p: There is at least one quadrilateral which has all it’s sides equal.
4. ~p is true. (square is a quadrilateral which has all it’s sides equal)

Part (iii):
1. Consider the statement:
p: The sum of 3 and 4 is 9.
2. The negation can be written as:
~p: It is false that the sum of 3 and 4 is 9.
OR
~p: sum of 3 and 4 is not 9.
3. ~p is true.


Compound statements

This can be explained in 4 steps:
1. Some mathematical statements are obtained by combining two or more simple statements.
• The simple statements are combined by using connecting words like “and”, “or” etc.,
2. This can be demonstrated using an example. It can be written in 2 steps:
(i) Consider the following statement:
p: There is something wrong with the bulb or with the wiring.
• This statement tells us that:
   ♦ The bulb is not turning on because,
   ♦ there is something wrong with the bulb.
   ♦ or there is something wrong with the wiring.
(ii) So the given statement is in fact formed by two statements:
q: There is something wrong with the bulb.
r: There is something wrong with the wiring.
   ♦ q and r are connected by “or”.
3. Another example:
(i) Consider the two statements:
p: 7 is an odd number.
q: 7 is a prime number.
• These two simple statements can be connected by the word “and”.
• We get:
r: 7 is both an odd number and a prime number.
4. A compound statement is a statement which is made up of two or more simple statements. Each of the simple statements is called a component statement.


Let us see some solved examples:
Solved example 14.3
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:
Part (i):
1. We have:
p: The sky is blue and the grass is green.
2. The component statements are:
q: The sky is blue.
r: The grass is green.
   ♦ q and r are connected by “and”.  

Part (ii):
1. We have:
p: It is raining and it is cold.
2. The component statements are:
q: It is raining.
r: It is cold.
   ♦ q and r are connected by “and”.  

Part (iii):
1. We have:
p: All rational numbers are real and all real numbers are complex.
2. The component statements are:
q: All rational numbers are real.
r:  All real numbers are complex.
   ♦ q and r are connected by “and”.

Part (iv):
1. We have:
p: 0 is a positive number or a negative number.
2. The component statements are:
q: 0 is a positive number.
r: 0 is a negative number.
   ♦ q and r are connected by “or”.

Solved example 14.4
Find the component statements of the following compound statements. Check each of the component statements and find whether they are true or false.
(i) A square is a quadrilateral and it’s 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:
Part (i):
1. We have:
p: A square is a quadrilateral and it’s four sides equal.
2. The component statements are:
q: A square is a quadrilateral.
r: A square has all four sides equal.
   ♦ q and r are connected by “and”.
3. Checking component statements:
• q is true.
• r is also true.

Part (ii):
1. We have:
p: All prime numbers are either even or odd
2. The component statements are:
q: All prime numbers are even numbers.
r: All prime numbers are odd numbers.
   ♦ q and r are connected by “and”.
3. Checking component statements:
• q is false.
• r is false.

Part (iii):
1. We have:
p: A person who has taken Mathematics or Computer science can go for MCA.
2. The component statements are:
q: A person who has taken Mathematics can go for MCA.
r: A person who has taken Computer science can go for MCA.
   ♦ q and r are connected by “or”.
3. Checking component statements:
• q is true.
• r is true.

Part (iv):
1. We have:
p: Chandigarh is the capital of Haryana and UP.
2. The component statements are:
q: Chandigarh is the capital of Haryana.
r: Chandigarh is the capital of UP.
   ♦ q and r are connected by “and”.
3. Checking component statements:
• q is true.
• r is false.

Part (v):
1. We have:
p: √2 is a rational number or an irrational number.
2. The component statements are:
q: √2 is a rational number.
r: √2 is an irrational number.
   ♦ q and r are connected by “or”.
3. Checking component statements:
• q is false.
• r is true.

Part (vi):
1. We have:
p: 24 is a multiple of 2, 4 and 8.
2. The component statements are:
q: 24 is a multiple of 2.
r: 24 is a multiple of 4.
s: 24 is a multiple of 8.
   ♦ q, r and s are connected by “and”.
3. Checking component statements:
q is true.
• r is true.
• s is true


• We see that:
Component statements are connected by words like “and”, “or” etc., These words have special meanings in mathematics. We will see it in the next section.


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

Exercise 14.2


In the next section, we will see special words/phrases.

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