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Unit-14(Examples)

Mathematical Reasoning

**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?

(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. (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.

**Example-2 :-** Write the negation of the following statements.

(i) Both the diagonals of a rectangle have the same length.

(ii) √7 is rational.

(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.

**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.

(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.

**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.

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’.

**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.

(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 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’

**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.

(i) The component statements are p: A line is straight. q: A line extends indefinitely in both directions. 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.

**Example-7 :-** For each of the following statements, determine whether an inclusive “Or” or exclusive “Or” is used. Give reasons for your answer.

(i) To enter a country, you need a passport or a voter registration card.

(ii) The school is closed if it is a holiday or a Sunday.

(iii) Two lines intersect at a point or are parallel.

(iv) Students can take French or Sanskrit as their third language.

(i) Here “Or” is inclusive since a person can have both a passport and a voter registration card to enter a country. (ii) Here also “Or” is inclusive since school is closed on holiday as well as on Sunday. (iii) Here “Or” is exclusive because it is not possible for two lines to intersect and parallel together. (iv) Here also “Or” is exclusive because a student cannot take both French and Sanskrit.

**Example-8 :-** Identify the type of “Or” used in the following statements and check whether the statements are true or false:

(i) √2 is a rational number or an irrational number.

(ii) To enter into a public library children need an identity card from the school or a letter from the school authorities.

(iii) A rectangle is a quadrilateral or a 5-sided polygon.

(i) The component statements are p: √2 is a rational number. q: √2 is an irrational number. Here, we know that the first statement is false and the second is true and “Or” is exclusive. Therefore, the compound statement is true.

(ii) The component statements are p: To get into a public library children need an identity card. q: To get into a public library children need a letter from the school authorities. Children can enter the library if they have either of the two, an identity card or the letter, as well as when they have both. Therefore, it is inclusive “Or” the compound statement is also true when children have both the card and the letter.

(iii) Here “Or” is exclusive. When we look at the component statements, we get that the statement is true.

**Example-9 :-** Write the contrapositive of the following statement:
(i) If a number is divisible by 9, then it is divisible by 3.

(ii) If you are born in India, then you are a citizen of India.

(iii) If a triangle is equilateral, it is isosceles.

The contrapositive of the these statements are (i) If a number is not divisible by 3, it is not divisible by 9. (ii) If you are not a citizen of India, then you were not born in India. (iii) If a triangle is not isosceles, then it is not equilateral. The above examples show the contrapositive of the statement if p, then q is “if ∼q, then ∼p”.

**Example-10 :-** Write the converse of the following statements.

(i) If a number n is even, then n² is even.

(ii) If you do all the exercises in the book, you get an A grade in the class.

(iii) If two integers a and b are such that a > b, then a – b is always a positive integer.

The converse of these statements are (i) If a number n² is even, then n is even. (ii) If you get an A grade in the class, then you have done all the exercises of the book. (iii) If two integers a and b are such that a – b is always a positive integer, then a > b.

**Example-11 :-** For each of the following compound statements, first identify the corresponding component statements. Then check whether the statements are true or not.

(i) If a triangle ABC is equilateral, then it is isosceles.

(ii) If a and b are integers, then ab is a rational number.

(i) The component statements are given by p : Triangle ABC is equilateral. q : Triangle ABC is Isosceles. Since an equilateral triangle is isosceles, we infer that the given compound statement is true.

(ii) The component statements are given by p : a and b are integers. q : ab is a rational number. Since the product of two integers is an integer and therefore a rational number, the compound statement is true.

**Example-12 :-** Given below are two pairs of statements. Combine these two statements using “if and only if ”.

(i) p: If a rectangle is a square, then all its four sides are equal.

q: If all the four sides of a rectangle are equal, then the rectangle is a square.

(ii) p: If the sum of digits of a number is divisible by 3, then the number is divisible by 3.

q: If a number is divisible by 3, then the sum of its digits is divisible by 3.

(i) A rectangle is a square if and only if all its four sides are equal. (ii) A number is divisible by 3 if and only if the sum of its digits is divisible by 3.

**Example-13 :-** Check whether the following statement is true or not. If x, y ∈ Z are such that x and y are odd, then xy is odd.

Let p : x, y ∈ Z such that x and y are odd q : xy is odd To check the validity of the given statement, we apply Case 1 of Rule 3. That is assume that if p is true, then q is true. p is true means that x and y are odd integers. Then x =2m + 1, for some integer m. y = 2n + 1, for some integer n. Thus xy = (2m + 1) (2n + 1) = 2(2mn + m + n) + 1 This shows that xy is odd. Therefore, the given statement is true.

**Example-14 :-** Check whether the following statement is true or false by proving its contrapositive. If x, y ∈ Ζ such that xy is odd, then both x and y are odd.

Let us name the statements as below p : xy is odd. q : both x and y are odd. We have to check whether the statement p ⇒ q is true or not, that is, by checking its contrapositive statement i.e., ∼q ⇒ ∼p Now ∼q : It is false that both x and y are odd. This implies that x (or y) is even. Then x = 2n for some integer n. Therefore, xy = 2ny for some integer n. This shows that xy is even. That is ∼p is true. Thus, we have shown that ∼q ⇒ ∼p and hence the given statement is true.

**Example-15 :-** Verify by the method of contradiction. p: √7 is irrational

Let √7 is a rational number. Therefore, we can find two integers a, b (b ≠ 0) such that √7 = a/b Let a and b have a common factor other than 1. Then we can divide them by the common factor, and assume that a and b are co-prime. a = b√7 squaring both sides a² = 7b² Therefore, a² is divisible by 7 and it can be said that a is divisible by 7. Let a = 7k, where k is an integer (7k)² = 7b² b² = 7k² This means that b² is divisible by 7 and hence, b is divisible by 7. This implies that a and b have 7 as a common factor. And this is a contradiction to the fact that a and b are co-prime. Hence, √7 cannot be expressed as p/q or it can be said that √7 is irrational.

**Example-16 :-** By giving a counter example, show that the following statement is false. If n is an odd integer, then n is prime.

The given statement is in the form “if p then q” we have to show that this is false. For this purpose we need to show that if p then ∼q. To show this we look for an odd integer n which is not a prime number. 9 is one such number. So n = 9 is a counter example. Thus, we conclude that the given statement is false.

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