ncert books for class maths cbsc

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 Write the negation of the following statements.

Both the diagonals of a rectangle have the same length. 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 is rational. This can also be rewritten as is not rational. MATHEMATICS Example Write the negation of the following statements and check whether the resulting statements are true, Australia is a continent.

There does not exist a quadrilateral which has all its sides equal. (iii) Every natural number is greater than . (iv) The sum of and is . 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. 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 . This can be rewritten as There exists a natural number which is not greater than . This is a false statement.

(iv) The negation is It is false that the sum of and is . This can be written as The sum of and is not equal to . This statement is true. .

. 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 MATHEMATICAL REASONING 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: is an odd number. q: is a prime number.

These two statements can be combined with “and” r: is both odd and prime number. This is a compound statement. This leads us to the following definition: Definition 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 Find the component statements of the following compound statements. The sky is blue and the grass is green. It is raining and it is cold.

(iii) All rational numbers are real and all real numbers are complex. (iv) 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 MATHEMATICS p: is a positive number.

q: is a negative number. The connecting word is ‘or’. Example Find the component statements of the following and check whether they are true or not. A square is a quadrilateral and its four sides equal.

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) is a rational number or an irrational number.

(vi) is a multiple of , and . 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’. 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 MATHEMATICAL REASONING p : is a rational number. q : 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: is a multiple of . q: is a multiple of . r: is a multiple of . 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 . . Write the negation of the following statements: Chennai is the capital of Tamil Nadu. is not a complex number (iii) All triangles are not equilateral triangle.

(iv) The number is greater than . (v) Every natural number is an integer. . Are the following pairs of statements negations of each other: The number x is not a rational number.

The number x is not an irrational number. The number x is a rational number. The number x is an irrational number. .

Find the component statements of the following compound statements and check whether they are true or false. Number is prime or it is odd. All integers are positive or negative. (iii) is divisible by , and .

. Special Words/Phrases Some of the connecting words which are found in compound statements like “And”, MATHEMATICS “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. . .

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: is divisible by , and . This statement has following component statements q: is divisible by .

r: is divisible by . s: is divisible by . Here, we know that the first is false while the other two are true. We have the following rules regarding the connective “And” .

The compound statement with ‘And’ is true if all its component statements are true. . 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 Write the component statements of the following compound statements and check whether the compound statement is true or false.

A line is straight and extends indefinitely in both directions. 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. MATHEMATICAL REASONING Both these statements are true, therefore, the compound statement is true. The component statements are p: is less than every positive integer. q: 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.