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Karnataka 1st PUC Maths Question Bank Chapter 14 Mathematical Reasoning
Question 1.
What is the basic unit involved in mathematical reasoning?
Answer:
The basic unit involved in mathematical statement.
Question 2.
Define mathematical statement.
Answer:
Mathematical statement is a sentence which, in a given context is either true or false but not both.
Question 3.
Define mathematically acceptable statements.
Answer:
A sentence is called mathematically acceptable statement if it either true or false but not both.
- Note: In this chapter, statement means it is mathematically acceptable statement.
- Note: If a sentence, which is vague or ambiguous then that sentence is not a statement.
- Note: Open sentences are not statements.
- Note: A sentence which contains where, why, what, he, she, other pronouns is not a statement.
Question 4.
Which of the following sentences are statements? Give reasons for your answers.
(i) Two plus two equals four.
(ii) The sum of two equals four
(iii) All prime numbers are add numbers
(iv) The sum of x and y is greater than zero
(v) How beautiful?
(vi) Open the door.
(vii) Where are you going?
(viii) Tomorrow is Friday
(ix) She is mathematic graduate.
(x) There are 40 days in a month.
(xi) 8 is less than 6.
(xii) Every set is a finite set.
(xiii) The sun is a star
(xiv) Mathematics is fun.
(xv) There is no rain without clouds.
(xvi) There are 35 days in a month.
(xvii) Mathematics is difficult.
(xviii) The sum of 5 and 7 is greater than 10
(xix) The square of a number is an even number.
(xx) The sides of a quadrilateral have equal length.
(xxi) Answer this question
(xxii) The product of (-1) and 8 is 8
(xxiii) Today is windy day.
(xxiv) All real numbers are complex numbers
(xxv) The sum of all interior angles of a triangle is 180°
Answer:
(i) It is a statement because it is always true.
(ii) It is a statement because it is always true.
(iii) It is a statement because it is false.
(vi) It is not a statement because it is open sentence.
(v) It is not a statement because it is an exclamation.
(vi) It is not a statement ∵ It is an order.
(vii) It is not a statement ∵ It is a question.
(viii) Not a statement, ∵ It is true on Thursday but not on other days.
(ix) Not a statement ∵ Sentence with variable pronoun like‘she’.
(x) A statement ∵ It is false (statement) sentence.
(xi) A statement ∵ It is a false sentence.
(xii) Not a statement ∵ It may be true or false
(xiii) A statement ∵ It is true
(xiv) Not a statement ∵ This sentence is not always true
(xv) A statement ∵ This sentence is always true, as is natural phenomenon.
(xvi) Not a statement ∵ It is false sentence maximum number of days in a month can never exceed.
(xvii) Not a statement ∵ It may be true or false for some people mathematics can be easy and some people mathematics can be difficult.
(xviii) A statement∵ It is true sentence. Since 5 + 7 = 12 >10
(xix) Not a statement, ∵ It is sometimes true and sometimes false. Since (2)2 = 4, even and (3)2 = 9, odd.
(xx) Not a statement ∵ It may be true or false.
Since square has equal length sides, rectangle has unequal length sides.
(xxi) Not a statement, ∵ It is an order.
(xxii) A statement ∵ It is false sentence.
(xxiii) Not a statement ∵ which day is not mentioned
(xxiv) A statement ∵ It is true sentence.
(xxv) A statement ∵ It is always true.
Question 5.
Define negation of a statement.
Answer:
If p is a statement, then the negation of p is also a statement and is denoted ~p and read as ‘not p’.
Note: While forming the negation of a statement, phrases like ‘It is not the case’ or ‘It is false that’ are also used.
Question 6.
Define compound statement.
Answer:
Compound statement is a statement which is made up of two or more statements using some connecting words like ‘and’ ‘or’, ‘if, then’, ‘if and only if etc.
Note: Connecting words like ‘And’ ‘Or’, ‘If, then’, ‘If and only if are called connectives and each statement is called a component statement.
Question 7.
Write the negation of the following statements:
(i) New Delhi is a city
(ii) Everyone in Germany speaks German
(iii) \(\sqrt{7}\) is rational
(iv) Chennai is the capital of Tamil Nadu
(v) \(\sqrt{2}\) is not a complex number
(vi) All triangles are not equilateral triangle
(vii) The number 2 is greater than 7.
(viii) Every natural number is an integer.
Answer:
(i) New Delhi is not a city.
(ii) It is false that everyone in Germany speaks German.
(iii) \(\sqrt{7}\) is not rational
(iv) Chennai is not the capital of Tamil Nadu
(v) \(\sqrt{2}\) is a complex number
(vi) The number 2 is not greater than 7
(vii) Every natural number is not an integer.
Question 8.
Are the following pairs of statements negations of each other,
(i) The number x is not a rational number The number x is not an irrational number,
(ii) The number x is a rational number. The number x is an irrational number
Answer:
(i) Let p : the number x is not a rational number
q : the number x is not an irrational number
Now ~ p : The number x is a rational number
q : The number x is a rational number (∵ when a number is not irrational, it is a rational)
~p = q and ~q = p
⇒ given pairs of statements are negations of each other.
(ii) Let p : the number x is a rational number
q : the number x is an irrational number. Now, ~p: the number x is an irrational number (∵When a number is not rational, it is irrational)
∴ ~p = q and ~q=p
⇒The pairs are negations of each other.
Question 9.
Find the component statements of the following compound statements and check whether they are true or false.
(i) Numbers 3 is prime or it is odd
(ii) All integers are positive or negative
(iii) 100 is divisible by 3,11 and 5
(iv) A square is a quadrilateral and its four sides equal
(v) All prime numbers are either even or odd
(vi) Chandigarh is the capital of Haryana and P.
(vii) 24 is a multiple of 2, 4 and 8
Answer:
(i) The component statements are
p : number 3 is prime
q : number 3 is odd
Both p and q are true
(ii) The component statements are
p : all integers are positive
q : all integers are negative
Both p and q are false
(iii) The component statements are
p : 100 is divisible by 3 (false)
q : 100 is divisible by 11 (false)
r : 100 is divisible by 5 (true)
(iv) The component statements are
p : a square is a quadrilateral
q : a square has all its sides equal
Both p and q are true
(v) The component statements are
p : all prime numbers are even
q : all prime numbers are odd
Both p and q are false
(vi) The component statements are
p : Chandigarh is the capital of Haryana (True)
q : Chandigarh is the capital of U.P. (False)
(vii) The component statements are
p : 24 is a multiple of 2 (True)
q : 24 is a multiple of 4 (True)
r : 24 is a multiple of 8 (True)
Question 10.
Write the rules for
(i) ‘And’ (conjunction) (∧)
(ii) ‘Or’ (disjunction) (∨)
(iii) ‘If-Then’ (implication) (⇒)
(iv) ‘If and only if (double implication) (⇔)
Answer:
Let p and q be the statements of the compound statement. Then
(i) The compound statement p and q is the conjunction of p and q and is denoted by
p ∧ q . (read: p and q)
Rule for compound statement with ‘And’ (conjunction)
This can be shown by the truth table
∴ Truth table for p ∧ q as follows
| p | q | p ∧ q |
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | F |
(ii) The compound statement p or q is the disjunction of p and q, and is denoted by p v q (read: p ∨ q)
Rule: The disjunction p ∨ q is false if both p and q are false, otherwise it is true.
This can be shown by truth table
| P | q | p ∧ q |
| T | T | T |
| T | F | T |
| F | T | T |
| F | F | F |
(iii) The compound statement ‘if p then q’ is implication of p and It is denoted by p → q or p ⇒ q. (read : p implication q)
Rule:
| P | q | p → q |
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
Note:
If ‘p’ and then ‘q’ is small following:
- p ⇒ q (i.e., p implies q)
- p is sufficient condition for q
- p only if q
- q is necessary condition for p
- ~q implies ~p (i.e., ~q ⇒ ~p)
- The compound statement ‘p if and only if q’ is double implication of p and It is denoted by p ⇔ q (read: p double implication q)
Rule:
| p | q | P ⇔ q |
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | T |
Note: ‘p if and only if q’ is same as the following.
- p ⇔ q
- p if and only if q
- q if and only if p
- p is necessary and sufficient condition for q and vice-versa
Question 11.
Define quantifiers (quantifiers)
Answer:
‘Quantifiers’ are the phrases like ‘there exists’ and ‘for all’ (or for all).
Question 12.
Define converse and contrapositive.
Answer:
Converse and contra positive are certain other statements which can be formed from a given statement with ‘if-then’.
- Converse: The converse of the statement if p, then q is defined as ‘if q then p’.
i.e., the converse of a implication p ⇒ q is q ⇒ q - Contrapositive: The contrapositive of the statement if p, then q is defined as ‘if ~q, then ~p’ i.e., the contrapositive of a implication p ⇒ q is ~q ⇒ ~p
Note:
Inverse of a implication p ⇒ q is ~p ⇒ ~q
Question 13.
For each of the following compounds 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 doesnot cool down fast at night.
(iv) x = 2 and x = 3 are the roots of the equation 3x2 – x -10 = 0
Answer:
(i) ‘and’ is the connecting word.
∴ component statements are
p : all rational numbers are real
q : all real numbers are not complex.
(ii) Connecting word is ‘or’ component statements are
p : square of an integer is positive
q : square of an integer is negative
(iii) Connecting word is ‘and’ component statements are
p : sand heats up quickly in the sun
q : sand does not cool down fast at night
(iv) Connecting word is ‘and’Component statements are
p : x – 2 is a root of 3x2 – x – 10 = 0
q : x = 3 is a root of 3x2 – x – 10 = 0
Question 14.
Identify the quantifier in the following statements and write the negation of the statement.
(i) There exists a number which is equal to its square
(ii) For every real number x, x is less than x + 1
(iii) There exists a capital for every state in India
Answer:
(i) Quantifier is‘there exists’
The negation is,
There doesn’t exist a number which is equal to its square.
(ii) Quantifier is ‘For every’
The negation is, there exist a real number x such that x is not less than x + 1
(iii) Quantifier is ‘there exists’
The negation state in India which doesn’t have a capital
Question 15.
Check whether the following pair of statements are negation of each other. Give reasons for your answer.
(i) x + y = y + x is true for every real numbers x and y
(ii) There exists real numbers x and y for which x + y = y + x
Answer:
No
Negation of (i) is ‘there exists real number x and y for which x + y & y + x’, which is not same as (ii).
Question 16.
State whether ‘or’ used in the following statements is exclusive or inclusive. Give reason for your answer.
(i) Sun rises or moon sets
(ii) To apply for a driving license, you should have a ration card or a pass port
(iii) All integers are positive or negative
Answer:
(i) Here ‘or’ is exclusive
∵ ‘sun rises’ and ‘moon sets’ cannot be true simultaneously
(ii) Here ‘or’ is inclusive
∵ For license, one can have both ration card and a passport.
(iii) Here ‘or’ is exclusive
∵ Integer cannot be both positive as well as negative.
Question 17.
Rewrite the following statements with ‘if-then’ in five different ways conveying the same meaning.
(i) ‘If a natural number is odd, then its square is also odd’
(ii) If a number is a multiple of 9, then it is a multiple of 3
Answer:
(i) Given: ‘If a natural number is odd, then its square is also odd.
Let p : a natural number is odd
q : a natural number square is also odd. Then, if p then q is same as
- p ⇒q (p implies q)
i.e., A natural number is odd implies that its square is also odd. - p is sufficient condition for q e., for the square of a natural number to be odd it is sufficient that the number itself is odd.
- p only if q e., A natural number is odd only if its square is odd.
- q is a necessary condition for p e., for a natural number to be odd it is necessary that its square must be odd.
- ~q implies ~p
- If the square of a natural number is not odd, then the number itself is also not odd.
(ii) Try yourself
Question 18.
Write the contrapositive and converse of the following statements.
(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, then it is isosceles
(iv) If a number n is even, then n2 is even
(v) If you do all the exercises in the book you get an A grade in the class.
(vi) If x is a prime number, then x is odd
(vii) If the two lines are parallel, then they donot intersect in the same plane
(viii) Something is cold implies that it has low temperature
(ix) x is even number implies that x is divisible by 4
Answer:
(i) Let p : A number is divisible by 9
q : A number is divisible by 3
Its converse is q ⇒ p and its contrapositive is ~q ⇒~P
∴ contrapositive: ‘If a number is not divisible by 3, then it is not divisible by 9’.
Converse: ‘If a number is divisible by 3 then it is divisible by 9.
(ii) Contrapositive: ‘If you are not a citizen of India then you were not born in India’. Converse: ‘If you are a citizen of India then you are born in India’.
(iii) Contrapositive: ‘If a triangle is not isosceles, then it is not equilateral’.
Converse: ‘If a triangle is isosceles, then it is equilateral’.
(iv) Try
(v) Try
(vi) Contrapositive: ‘If x is not odd, then x is not a prime number’.
Converse: If x is odd then x is a prime number
(vii) Try
(viii) and (ix) yourself
Question 19.
Write each of the following statements in the form ‘if-then’
(i) You get a job implies that your credentials are good.
(ii) The Banana trees will bloom if it stays warm for a month
(iii) A quadrilateral is a parallelogram if its diagonals bisect each other
(iv) To get an A+ in the class, it is necessary that you do all the exercise of the book
Answer:
(i) ‘If you get a job, then your credentials are good’.
(ii) ‘If the banana tree stays warm for a month, then it will bloom’
(iii) ‘If the diagonals of a quadrilateral bisect each other, then it is a parallelogram’.
(iv) ‘If you get an A+ in the class then you have done all exercises of the book’.
Question 20.
Verify by the method of contradiction that \(\sqrt{7}\) is irrational
Answer :
Let p : \(\sqrt{7}\) is irrational
Let us assume p is not true i.e.., \(\sqrt{7}\) is rational .
⇒ \(\sqrt{7}=\frac{a}{b}\),where a and b are integers having no common
factor.
⇒ \(7=\frac{a^{2}}{b^{2}}\)
⇒ a2 =7b2
⇒ 7 divides a2
⇒ 7 divides a
⇒ a = 7c, for some integer c.
⇒ a2 = 49c2
⇒ 7b2 = 49c2
⇒ b2 = 7c2
⇒ 7 divides b2
⇒ 7 divides b
Thus, 7 is common factor of both a and b. This contradicts that a and b have no common factor. So, our assumption is wrong. Hence, \(\sqrt{7}\) is irrational is true.