MATHEMATICAL REASONING

1. There are two types of reasoning the deductive and inductive. Deductive reasoning was developed by Aristotle, Thales, Pythagoras in the classical Period (600 to 300 B.C.).

2. In deduction, given a statement to be proven, often called a conjecture or a theorem, valid deductive steps are derived and a proof may or may not be established. Deduction is the application of a general case to a particular case.

3. Inductive reasoning depends on working with each case, and developing a conjecture by observing incidence till each and every case is observed.

4. Deductive approach is known as the top-down" approach”. Given the theorem which is narrowed down to specific hypotheses then to observation. Finally the hypotheses is tested with specific data to get the confirmation (or not) of original theory.

5. Mathematical reasoning is based on deductive reasoning.

The classic example of deductive reasoning, given by Aristotle, is

• All men are mortal.

• Socrates is a man.

• Socrates is mortal.

6. The basic unit involved in reasoning is mathematical statement.

7. A sentence is called a mathematically acceptable statement if it is either true or false but not both. A sentence which is both true and false simultaneously is called a paradox.

8. Sentences which involve tomorrow, yesterday, here, there etc i.e variables etc are not statements.

9. The sentence expresses a request, a command or is simply a question are not statements.

10. The denial of a statement is called the negation of the statement. While forming the negation of a statement, phrases like , “It is not the case” or “It is false that” are used.

11. Two or more statements joined by words like “and” “or” are called Compound statements. Each statement is called a component statement. “and” “or” are connecting words.

12. An “And” statement is true if each of the component statement is true and it is false even if one component statement is false.

13. An “OR” statement is will be true when even one of its components is true and is false only when all its components are false.

14. The word “OR” can be used in two ways (i) Inclusive OR (ii) Exclusive OR. If only one of the two options is possible then the OR used is Exclusive OR. If any one of the two options or both the options are possible then the OR used is Inclusive OR.

15. There exists “∃” and “For all” ∀ are called quantifiers.

16. A statement with quantifier “There exists” is true, if it is true for at least one case.

17. If p and q are two statements then a statement of the form 'If p then q' is known as a conditional statement. In symbolic form p implies q is denoted by p ⇒ q.

18. The conditional statement p ⇒ q can be expressed in the various other forms:

(i) q if p (ii) p only if q (iii) p is sufficient for q (iv) q is necessary for p.

19. A statement formed by the combination of two statements of the form if p then q and if q then p is p if and only if q. It is called biconditional statement.

20. Contrapositive and converse can be obtained by a if then statement The contrapositive of a statement

p ⇒ q is the statement ∼ q ⇒ ∼p The converse of a statement p ⇒ q is the statement q ⇒ p

21. T values of various statement:

p

q

p and q

p or q

p⇒q

T

T

T T T

T

F

F T F

F

T

F T T

F

F

F F T

22. Two prove the truth of an if p then q statement. There are two ways: the first is assume p is true and prove q is true. This is called the direct method. Or assume that q is false and prove p is false. This is called the Contrapositive method.

23. To prove the truth of “ p if and only if q” statement , we must prove two things , one that the truth of p implies the truth of q and the second that the truth of q implies the truth of p.

24. The following methods are used to check the validity of statements: (i) Direct method (ii) Contrapositive method (iii) Method of contradiction (iv) Using a counter example

25. To check whether a statement p is true, we assume that it is not true, i.e. ∼p is true. Then we arrive at some result which contradicts our assumption.

EXAMPLES :

1. Check whether the following sentences are statements(or composition). Give reasons for your answer. (i) Every set is finite set. (ii) The sun is a star. (iii) There is no rain without clouds. (iv) How far is Chennai from here?Answer: (i) It is statement ∵this sentence is false since there are sets which are not finite. (ii) It is statement ∵ it is scientifically established fact (iii) same as (ii) part. (iv) not statement ∵ this is question which contain word “here”. 2. Write the negation of the following statements(i) Both the diagonals of a rectangle have the same length.(ii) √ 7 is rational.(iii) Every one in Germany speaks German.(iv) There does not exist a quadrilateral which has all it’s sides equal.Answer: (i)It is false that both the diagonals in a rectangle have the same length. (ii) √ 7 is not rational.(iii)Not every one in Germany speaks German. It says merely that atleast one person in Germany does not speak German.(iv) It is not the case that there does not exist a quadrilateral which has all it’s sides equal. This also means there exists a quadrilateral which has all it’s sides equal. 3. Find the compound statements of the following and check whether they are true or not.(i) All prime numbers are either even or odd.(ii) 24 is a multiple of 2,4 and 8Answer: (i) compound 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’. (ii) p : 24 is a multiple of 2. q: 24 is a multiple of 4. r: 24 is a multiple of 8. All three statements are true , connecting word is `and’.

4. For each of the following statements, determine whether an inclusive “Or’’ or exclusive “Or” is used. Give reasons for your answer.(i) The school is closed if it is a holiday or a Sunday. (ii) Two lines intersect at a point or are parallel.Answer: (i) Here “Or” is inclusive since school is closed on holiday as well as on Sunday. (ii) Here “Or” is exclusive because it’s not possible for two lines to intersect and parallel together.5. Write the contrapositive and converse of statement If a triangle is equilateral, it is isosceles.Answer: contrapositive is If a triangle is not isosceles, then it is not equilateral. Converse is If a triangle is isosceles, then it is equilateral. 6. Combine two given statements using “if and only if” . p: If the sum of digits of a number is divisible by 3, the number is divisible by 3. q: If a number is divisible by 3, then the sum of its digits is divisible by 3. Answer: A number is divisible by 3 if and only if the sum of its digits is divisible by 3.7. 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.Solution: Let p: x, y∊ Z are such that x and y are odd q: xy is odd we apply direct method(if p is true, then q is true) of Rule3( statements with “If – 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 ⇨ odd ∴ statement is true. We can check it by using case 2(contrapositive) of Rule 3 ~ q : product xy is even.This is possible only if either x or y is even. This shows that p is not true. Thus we have shown that ~q ⇨ ~p 8. By giving a counter example (example of situationwhere the statement is not valid), show that the following statement is false.

If n is an odd integer, then n is prime.Solution: Statement is in the form of “if p then q” we need to show if p then ~q. n=9 is a counter example which is odd but not prime, we conclude that the given statement is false. 9. (misc.) Write the negation of the following statements:(i) p: For every real number x, x2>x. (ii) q: There exists a rational number x such that x2 = 2.(iii) r: All birds have wings.(iv) s: All students study mathematics at the elementary level.Answer: (i) ~p: There exists a real number x, such that x2< x. (ii) ~ q: For all real numbers x, x2 ≠ 2. (iii) ~r: There exists a bird which have no wings. (iv) ~s: (iv) s: There exists a student who does not study mathematics at the elementary level.10. Using the words “necessary and sufficient” rewrite the statement “The integer n is odd if and only if n2 is odd”.Also check whether the statement is true.Answer: The necessary and sufficient condition that the integer n is odd is n2 must be odd p: the integer n is odd. q: n2 is odd. We have to check whether “if p then q” and “if q then p”is true.Case.1 If p, then q Let n=2k+1 , k is an integer. n2 = (2k+1)2 odd case.2 If q, then p We check this by contrapositive method. Means if ~p , then ~q ~p: n is even ⇨ n = 2k for some k. Then n2 = 4k2 (even).11.Write the component statements of the following compound statement “All prime numbers are either even or odd”.Answer: The component statements are P: All prime numbers are even number. q: All prime numbers are odd number. Both these statements are false and the connecting word is `or’. **Biconditional of statement:

P: Today is 14th of August. q: Tomorrow is Independence day. Is p ↔ q is given by “Today is 14th of August iff tomorrow is Independence day”.

Questions(NCERT)

Question1: Give three examples of sentences which are not statements. Give reasons for the answer.

(i) Do your duty. [∵ it is an order.]

(ii) How is your friend? [∵ it is an interrogative type]

(iii) How beautiful ! [∵ it is an exclamation]

Question2: Are the following pairs of statements negations of each other?

(i) The number x is not a rational number. (YES)

The number x is not an irrational number.

(ii) The number x is a rational number. (YES)

The number x is an irrational number.

Answer: (i) yes, p: The number x is not a rational number.

~p: The number x is a rational number.(Same as second statement)

Question :

For each of the following compound statements first identify the connecting words and then break it into component statements.

(i) All rational numbers are real and all real numbers are not complex.

(ii) Square of an integer is positive or negative.

(iii) The sand heats up quickly in the Sun and does not cool down fast at night.

(iv) x = 2 and x = 3 are the roots of the equation 3x2 – x – 10 = 0.

Question :

Check whether the following pair of statements is negation of each other. Give reasons for the answer.

(i) x + y = y + x is true for every real numbers x and y.

(ii) There exists real number x and y for which x + y = y + x. [NO]

Answer: Negation of (i) is x + y = y + x is not true for every real numbers x and y.

Negation of (ii) is There does not exist real numbers x and y for which x + y = y + x.

Question :

Show that the statement

p: “If x is a real number such that x3 + 4x = 0, then x is 0” is true by

(i) direct method

(ii) method of contradiction

(iii) method of contrapositive

Solution: Let q: x is a real no. such that x3+4x=0. r: x is 0. , then, p: if q,then r .

(i) direct method: q is true ⇨ x(x2+4) = 0 ⇨x=0 [∵ x is real], so r is true, hence p is true.

(ii) method of contradiction: Let p be not true ⇨ ~p is true

⇨ ~(q ⇨ r) [∵p:q⇨r] ⇨ q and ~r is true [∵~(q ⇨ r)= q and ~r]⇨ x is a real number such that x3+4x=0 and x≠0⇨ x=0 and x≠0.

(iii) method of contrapositive: Let r be not true. Then,

R is not true ⇨ x≠0, x∊ R ⇨ x(x2+4)≠0, x∊R ⇨ q is not true.

Thus, ~r ⇨ ~q. hence p: q⇨ r is true.Question :

Show that the following statement is true by the method of contrapositive.

p: If x is an integer and x2 is even, then x is also even.

Solution: p: x is an integer and x2 is even , q: x is even.

Let q is false ∴ ~q is true , ∴ x is odd integer, let x=2k+1, x2 = (2k+1)2

∴ p is false. Thus ~ q ⇨ ~p.Question :

Which of the following statements are true and which are false? In each case give a valid reason for saying so.

(i) p: Each radius of a circle is a chord of the circle.

(ii) q: The centre of a circle bisects each chord of the circle.

(iii) r: Circle is a particular case of an ellipse.

(iv) s: If x and y are integers such that x > y, then –x < –y.

(v) t: is a rational number.

Answer: (i) F (ii) F [ a chord may or may not pass thro. The centre of the circle] (iii) T [ when major axis = minor axis](iv) T (V) F [contrad.]

Question :

State the converse, inverse and contrapositive of each of the following statements:

(i) p: A positive integer is prime only if it has no divisors other than 1 and itself.

(ii) q: I go to beach whenever it is a sunny day.

(iii) r: If it is hot outside, then you feel thirsty.

(iv) s: If x < y, then x2<y2.

Answer: (i) Converse: If a positive integer has no divisors other than 1 and itself, then it is prime.

Inverse: If a positive integer is not prime, then it has divisors other than 1 and itself.

Contrapositive: If a positive integer has divisors other than 1 and itself, then it is not a prime.

(ii) Converse: If I go to beach, then it is a sunny day.

Inverse: If it is not a sunny day, then I will not go to beach.

Contrapositive: If I do not go to beach, then it is not a sunny day.

(iii) Converse: If you feel thirsty, then it is hot outside.

Inverse: If it is not hot outside, then you do not feel thirsty.

Contrapositive: If you do not feel thirsty, then it is not hot outside.

(iv) Converse: If x2 < y2 , then x < y.

Inverse: If x is not less than y , then x2 not less than y2 .

Contrapositive: If x2 not less than y2 , then x is not less than y.

Question :

Re write each of the following statements in the form “p if and only if q”.

(i) p: If you watch television, then your mind is free and if your mind is free, then you watch television.

(ii) q: For you to get an A grade, it is necessary and sufficient that you do all the homework regularly.

(iii) r: If a quadrilateral is equiangular, then it is a rectangle and if a quadrilateral is a rectangle, then it is equiangular.

Solution: (i) You watch television if and only if your mind is free.

(ii) You will get an A grade if and only if you do all the homework regularly.

(iii) A quad. Is equi. If and only if it is a rectangle.

Question :

Check the validity of the statements given below by the method given against it.

(i) p: The sum of an irrational number and a rational number is irrational (by contradiction method).

(ii) q: If n is a real number with n > 3, then n2 > 9 (by contradiction method).

Answer: (i) Let statement be false i.e., irrat.+rat.number is rational.

An irrational no. = rational no. – rational no. = rational no.

Not possible ∴ we arrive at contradiction ∴ statement is wrong

(ii) Let if possible n2 > 9 is false ∴ n2 ≤ 9⇨ n2 - 9≤0 ⇨ (n-3)(n+3)≤0⇨ n+3≤0 [n > 3] , not possible ∴supposition is wrong, hence true.

Question :

Write the following statement in five different ways, conveying the same meaning.

p: If triangle is equiangular, then it is an obtuse angled triangle.

Answer: Let p: triangle is equiangular.

q: it is an obtuse angled triangle.

(i) p ⇨ q (ii) p is sufficient condition for q (iii) p only if q (iv) q is necessary condition for p (v) ~q⇨ ~p

Question: Identify the quantifier in the following statements and write the negation of the statements.

(i) There exists a number which is equal to its square.

The negation is There does not exist a number which is equal to its square. The quantifier is (“There exists”)

(ii) For every real number x, x is less than x+1. (“For every”)

The negation is There exists a real number x, x is not less than x+1.

(iii) There exists a capital for every state in India. (“There exists”)

The negation is

There exists a state in which does not have a capital.

ASSIGNMENT

Question1 Prove by using direct method and contrapositive method that the following statement is true.

P: For any reals x, y if x = y then 2x+a = 2y+a when a∊Z.

Question2 Check validity of the statements:

(i) p: 100 is multiple of 4 and 5 (ii) q : 60 is multiple of 3 or 5.

Question3 Let S be non- empty subset of R. Consider the following statement: p: There is a rational number x∊S s.t. x>0.

Which of the following statements is the negation of statement p?

(1) There is no rational no. x∊S s.t. x≤0.

(2) Every rational no. x∊S satisfies x≤0.

(3) There is a rational no. x∊S s.t. x≤0.

Question4 Write down negation of following:

(i) All real nos. Are rationals or irrational.

(ii) 35 is a prime no. or a composite no.

(iii) |x| is equal to either x or –x.

Question5 Re write each of the following statements in the form of conditional statements

(i) The unit digit of an integer is o or 5 if it is divisible by 5.

(ii) The square of a prime no. is not prime.

(iii) 2b = a+c, if a,b and c are in A.P.

Question6 Form the biconditional statement p ↔ q , where

(i) p: A natural no. n is odd. q: Natural no. n is not divisible by 2.

(ii) p: A triangle is an equi. Triangle. q: All three sides of a triangle are equal.

Question7 Write contrapositive

(i) If x=y and y=3, then x=3.

(ii) If natural no. n is divisible by 6, then n is divisible by 2 and 3.

(iii) If x is a real no. such that 0<x<1, then x2 <1.