Mathematics MATHEMATICAL REASONING - Negation of a statement and Compound statements
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### Topics Covered

star Negation of a statement
star Compound statements

### Negation of a statement

\color{fuchsia}{★ul"Negation of a statement")

color(green)("The denial of a statement is called the negation of the statement.")

If p is a statement, then the negation of p is also a statement and is color(green)("denoted by ∼ p,") and read as "‘not p’."

color(red)("Let us consider the statement "):

\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ color(blue)"p: New Delhi is a city"

The negation of this statement is

\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ "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

\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ color(orange)"New Delhi is not a city."

\color{green} ✍️ \color{green} \mathbf(KEY \ CONCEPT)

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.

color(red)("Let us consider the statement")

\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ color(orange)" 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 color(orange)"at least one person in Germany does not speak German."
Q 3161601525

Write the negation of the following statements.
(i) Both the diagonals of a rectangle have the same length.
sqrt7 is rational

Solution:

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 sqrt7 is rational
This can also be rewritten as
sqrt7 is not rational.
Q 3171601526

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:

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 quadrilaterial 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.

### Compound statements

\color{fuchsia}{★ul"Compound statements")

color(green)"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."

Many mathematical statements are obtained by combining one or more statements using some connecting words like "“and”, “or”," etc.

color(red)("Consider the following statement")

\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ color(orange)" 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 something wrong with the wiring. That means the given statement is actually made up of two smaller statements:

\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \color(orange) "q: There is something wrong with the bulb."

\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \color(orange) "r: There is something wrong with the wiring." connected by "“or”"

color(red)("Now, suppose two statements are given as below :")

\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ color(orange) " p: 7 is an odd number."

\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \color(orange) " q: 7 is a prime number."

These two statements can be combined with “and”

\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ color(orange) " r: 7 is both odd and prime number."

color(blue)("This is a compound statement.")

Q 3101601528

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
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ "p: 0 is a positive number."
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ "q: 0 is a negative number."
The connecting word is ‘or’.
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ "p: 0 is a positive number".
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ "q: 0 is a negative number."
The connecting word is ‘or’.
Q 3121701621

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) sqrt2 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 number."
 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \"q: All prime numbers are even number."
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
 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \p: sqrt2 "is a rational number."
 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \q: sqrt2 "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 matter in the following section.