+2 1st Year Elements of Mathematics Chapter 1 Mathematical Reasoning Questions & Answers

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CHAPTER 1 MATHEMATICAL REASONING

1. CONCEPT MAP

Proposition : (Mathematically acceptable)

A proposition (or Mathematically acceptable statement) is a declarative sentence which is either true or false but not both.

Thus a sentence will be a statement if

(a) It is declarative

(b) It has a truth value (either true (T) or false (F))

A sentence cannot be statement if it is

(a) A question

(b) An order

(d) A wish

(e) An advice or it involves

(i) Variable time like 'today', 3. 'tomorrow' 'yesterday' etc.

(ii) Variable place like 'here', 'there' etc.

(iii) Pronouns like 'he', 'she', 'they' etc.

(iv) Relative words / adjectives / undefined words like 'good', 'bad', 'beautiful', 'wise', etc.

(v) Variables x, y, z, u, v... etc.

We denote statements by small letters like p, q, r, s, etc.

2. Negative (~):

Denial of a statement is its negation.

Axiom of negation.

For any proposition p, if p is true, then ~p (Negation of p) is false and if p is false, then ~ p is true,

Truth table of Negation

p ~p

T F

F T

Logical Connectives

Two statements can be combined together by using the words like or, and, if, only if, if and only if etc. These are known as logical connectives.

A proposition in which one or more connectives appear, is known as composite or compound proposition.

4. Conjuction ( Ʌ ), (and):

Axiom: A conjuction p Ʌ q is true if both 'p' and 'q' are both true and false otherwise.

Truth table :

p q p Ʌ q

T T T

T f f

f T f

f f f

5. Disjunction (v) (or):

Asiom: A disjunction pvq is false only when both 'p' and 'q' are false and is true otherwise.

Truth table:

p q p v q

T T F

T F T

F T T

F F F

Inclusive and exclusive sence of 'OR'

An employee either goes on leave or attends to his duty. (Exclusive)

In this restaurant you can order for veg or non-veg items. (Inclusive)

6. Conditional (→) (if ... then):

Axiom: A conditional p→ q is false only when 'p' is true and 'q' is false in all other cases it is true.

Truth table:

p q p→ q

T T T

T F F

F T T

F F T

7. Converse, inverse and contrapositives

Converse of p→q is q→ p

Inverse of p→ q is ~p→~q

Contra positive of p→ q is ~q → ~p

8. Biconditional (p↔ q) (p if and only if q):

Axiom A biconditional p↔ q is true if both 'p' and 'q' have same truth value and is false otherwise.

Truth table

p q p↔ q

T T T

T f f

f T f

f f T

9. Equivalent statements :

Two statements 'p' and 'q' are said to be equivalent statements if they have same truth values.

10. Tautology:

A statement is a tautology if it is always true

11. Implication and double implication :

If a conditional p→ q is a tautology then we say p implies q and we write p⇒q.

If a biconditional p↔ q is a tautology and we write p⇒ q.

12. Contradiction:

A contradiction we mean a proposition which is false for all possible assignments of truth values to its prime components.

13. Logical quantifiers:

Logical quantifiers are the words which associates a quantity to it. There are two types of logical quantifiers.

(i) Existential (There exists)

(ii) Universal (For all, for every).

Mathematical Reasoning Exercise 1(a)

1. Choose the correct answer from the given choices:

(i) Negation of ‘Paris is in France and London is in England’ is :

(A) Paris is in England and London is in France.

(B) Paris is not in France or Landon is not in England.

(D) Paris is not in France and London is not in England.

(ii) The conditional (p ∧ q) → p is :

(A) a tautology

(B) a contradiction

(D) none of these

(iii) Which of the following is a contradiction?

(A) (p ∧ q) ∧ ~ (p ∧ q)

(B) p ∨ (~ p ∧ q)

(D) None of these

(iv) Which of the following is logically equivalent to ~ (~ p → q)?

(A) p ∧ q

(B) p ∧ ~ q

(D) ~ p ∧ ~ q

(v) (~(~p) ∧ q) is equivalent to

(A) ~ p ∧ q

(B) p ∧ q

(D) ~ p ∧ ~ q

vi) If p: It rains today, q: I go to school, r: I shall meet any friend, and s’: I shall go to a movie, then which of the following is the proposition.

If it does not rain or if I do not go to school, then I shall meet a friend and go to a movie.

(A) ∼ (p ∧ q) → (r ∧ s)

(B) ~ (p ∧ q) → (r ∧ s)

(D) None of these

(vii) Which of the following is true?

(A) p → q ≡ (~p → ~q)

(B) ~ (p → ~ q) ≡ (~ p ∧ q)

(C) ~ (~p ~ q) ≡ (~ p ∧ q)

(D) ~ (p ↔ q) ≡ [~(p → q)∧~(q → p)]

(viii)The Inverse of the proposition p ∧ ~ q) → r is:

(A) ~ r (~ p ∨ q)

(B) ~ p ∨ q → ~ r

(D) None of these

(ix) The contrapositive of (p ∨ q) → r is

(A) r → (p ∨ q)

(B ) ~r → (p ∨ q)

(C) ~ r → (~p ∧ ~q)

(D) p → (q ∨ r)

(x) Which of the following is inverse of the proposition: “If a number is a prime, then it is odd.

(A) if a number is not prime, then it is odd.

(B) if a number is not a prime, then it is not odd.

(D) if a number is not odd, then it is a prime.

2. Give examples, five in each case, of , sentences that are

(i) propositions

Ans:

Examples of a proposition:

(a) Rama is a boy.

(b) He is ten years old.

(d) He has 2 brothers and 3 sisters.

(e) 2 + 3 ≥ 5

(ii) not-propositions

Ans;

Examples of not-proposition:

(a) How old are you?

(b) What is he to you?

(d) What a beautiful girl!

(e) May God offer you long life.

3. Illustrate the use of all connectives and the modified ‘not’ in five separate examples of propositions.

Ans:

Conjunction: Hari and Madhu are two friends.

Disjunction: Hari or Madhu stand first in their class.

Implication: If I will be the king, then I will make you my queen.

Double Implication: I will love you if you will love me.

Negation: I don’t like her.

4. Try to construct an example of a proposition involving all connectives and also the modifier ‘not’.

Ans.

If she do not come and take my help, then she neither understands the problem nor solves them but she can create other problems.

5 . Which of the following sentences are propositions and which are not? Write with reasons :

(i) 2 < 5.

Ans. 2 < 5 is a proposition, as it is true.

(ii) Is 9 < 3?

Ans. Is 9 < 3? Is not a proposition, as it is neither true nor false.

(iii) x is greater than 100.

Ans. “x is greater than 100” is not a statement as x is not defined.

(iv) Why are you crying?

Ans. Why are you crying? Is not a statement as it is neither true nor false.

(v) May God grant you long life.

Ans. “May God grant you long life” is not a statement as it is neither true nor false.

(vi) Cuttack is a big city.

Ans. “Cuttack is a big city” is not a statement, as it contains the word ‘big’ (not defined).

(vii) It is possible that there is life on Mars.

Ans. “It is possible that there is life on Mars” is not a proposition. (To date it is not confirmed)

(viii)Ram is a friend of Hari.

Ans. Ram is a friend of Hari. It is a statement as it is true or false.

(ix) x2 – x + 1=0

Ans. x2 – x + 1 = 0” is not a statement as ‘x’ is not defined.

(x) Oh! What scenery!

Ans. Oh! What scenery! It is not a statement as it is neither true nor false.

(xi) You must go to school every day.

Ans. You must go to school every day. It is not a statement as it is neither true nor false.

(xii) It was raining yesterday.

Ans. It was raining yesterday. It is not a proposition as it is not known to which day yesterday means.

6. Write down negations of each of the following.

(i) If you read, you will pass.

Ans. It is not true that if you read, you will pass.

(ii) John is a friend of Thomas.

Ans. John is not a friend of Thomas.

(iii) Fifteen is greater than five.

Ans. Fifteen is not greater than five.

(iv) Either Pramod is clever or he is laborious.

Ans. Pramod is not clever and also not laborious.

(v) Money is necessary for happiness.

Ans. Money is not necessary for happiness.

(vi) It is raining and Mahanadi is flooded.

Ans. It is neither raining nor Mahanadi is flooded.

(vii) Pen is might than a sword.

Ans . Pen is not more mighty than a sword.

(viii) |x| is equal to either x or – x.

Ans. |x| is neither x nor – x

(ix) It is raining and it is cool.

Ans. It is not raining or it is not cool.

(x) 3 + 6 > 8 and 2 + 3 < 6.

Ans. 3 + 6 ≤ 8 or 2+ 3 ≥ 6.

7. Translate the following propositions into symbolic form, stating the prime components in each case.

(i) If you do not work hard, you will repent.

Ans:

(i) Let p: You work hard,

q: you will repent.

Then the given statement is ~ p → q.

(ii) Jamini will be rewarded if and only if he is punctual.

Ans: (ii) Let p: Jamini will be rewarded,

q: He is punctual.

The given statement is p ↔ q.

(iii) If there is a will, there is a way.

Ans: Let p: There is a will,

q: There is a way.

∴ The given statement is p → q.

(iv) Time and tide wait for none.

Ans. Let p: Time waits for none.

q: Tide waits for none.

∴ Solution is p ∧ q.

(v) 29 is a prime number which is a sum of two squares.

Ans. Let p: 29 be a prime number.

q: It is a sum erf two squares

∴ Solution is p ∧ q.

(vi) Life is short, but virtue is lasting.

Ans: Let p: Life is short.

q: Virtue is lasting.

Solution is q ∧ q.

(vii) If the boy is poor, then he will be hungry and if he is hungry, then he cannot be honest.

Ans: Let p: The boy is poor.

q: He will be hungry.

r: He can be honest.

Solution is (p → q) ∧ (q → ~r)

(viii) A year consists of twelve months while a month does not consist of more than thirty-one days.

Ans: Let p: A year consists of twelve 9. months.

q: A month consists of more than 31 days.

∴ Solution is p ∧ ~ q.

(ix) If the government cannot solve the unemployment problem, then public opinion will rise against it which will lead to a strengthening of opposition.

Ans; Let p: The Government can solve the unemployment problem.

q: Public opinion will rise against it.

r: It will lead to a strengthening of opposition.

∴ Solution is (p → q) → r.

(x) Chinu and Minu went to Calcutta, but Minu came back earlier since she lost all her money

Ans: Let p: Chinu went to Calcutta.

q: Minu went to Calcutta.

r: Minu came back earlier.

s: She lost all her money.

Solution is (p ∧ q) ∧ (s → r).

8. Let p, q, and r denote respectively the statements: “you are honest”, “you are laborious” and “you will receive a promotion”, Translate the following statements into English language :

(i) (p ∨ q) → r
(ii) ~ r → ~ p
(iii) ~ (p ∨ q) → ~ r
(iv) |r ∨ (~ p)| → ~ q
(v) p ∧ q ∧ r

Solution:

Let p: You are honest. q: You are laborious. r: You will receive a promotion.

(i) (p ∨ q) → r
Ans: (p ∨ q) → r. It’s the English language is “If you are honest or laborious then you will receive a promotion.”

(ii) ~ r → ~ p
Ans: If you will not receive a promotion, then you are not honest.

(iii) ~ (p ∨ q) → ~ r
Ans: If you are neither honest nor laborious then you will not receive a promotion.

(iv) |r ∨ (~ p)| → ~ q
Ans: If you will receive a promotion or you are not honest then you are not, laborious.

(v) p ∧ q ∧ r

Ans: You are honest and laborious and you will receive a promotion.

9.Construct truth tables for the following and indicate which of these are tautologies :

(i) p ∧ q → p ∨ q

p	q	p ∧ q	p ∨ q	p ∧ q ∧ p ∨ q
T	T	T	T	T
T	F	F	T	T
F	T	F	T	T
F	F	F	F	T

∴ It is a tautology.

(ii) p ∧ q → p

p	q	p ∧ q	p ∧ q → p
T	T	T	T
T	F	F	T
F	T	F	T
F	F	F	T

∴ It is a tautology.

(iii) p ∧ (p → q) → q

p	q	p → q	p ∧ (p → q)	p ∧ (p → q) → q
T	T	T	T	T
F	T	T	F	T
T	F	F	F	T
F	F	T	F	T

∴ It is a tautology.

(iv) p → p ∧ q

p	q	p ∧ q	p → (p ∧ q)
T	T	T	T
T	F	F	F
F	T	F	T
F	F	F	T

∴ It is a not tautology.

(v) p → (~p → q)

p	q	~p	~p → q	p → (~p → q)
T	T	F	T	T
T	F	F	T	T
F	T	T	T	T
F	F	T	F	T

∴ It is a tautology.

(vi) ~p ∧ (p ∧ q) → q

(vii) (p ∨ ~q) ∧ (q ∨ ~p)

(viii) p → (~ q ∧ r)

(ix) (p → q) → [(q → r) → (p → r)]

(x) p ∨ → ~(p ∧ q)

(xi) (p → ~p) → ~p

(xii) (~ p ∨ p) → (~ q ∨ q)

(xiii) (p ∧ q) → p) → q

(xiv) (p ↔ q) ∧ (q ↔ r) → (p ↔ r)

(xv) [p → (p ∨ q)] → [q → (p ∧ q)]

Question 10.

If p has truth value T, what can be said about the truth values of

(i) ~p ∧ q → p ∨ q

(ii) p ∨ q → ~p ∧ q

Question 11.
Determine the truth values of p ↔ ~ q when the biconditional p ↔ q has truth value (i) F, (ii) T.
Solution:

p	q	-q	p ↔ q	p ↔ ~q
T	T	F	T	F
T	F	T	F	T
F	T	F	F	T
F	F	T	T	F

(i) When p ↔ q is False, p ↔ ~ q is True.
(ii) ∴ When p ↔ q is True, p ↔ ~q is False.

12. State the converse, inverse and contra-positive of each of the following propositions. Stating it as a conditional, wherever necessary.

(i) If A ABC is equilateral, then its three angles are congruent.
Ans:
If A ABC is equilateral, then its three angles are congruent.

Converse: If the three angles of the A ABC are congruent, then it is equilateral.

Inverse: If A ABC is not equilateral, then its three angles are not congruent.

Contrapositive: If the three angles of the A ABC are not congruent, then it is not equilateral.

(ii) If Gopal is clever, then he is rich.
Ans:
Con: If Gopal is rich, then he is clever.
Inv.: If Gopal is not clever, then he is not rich.
Cont: If Gopal is not rich, then he is not clever.

(iii) p → ~ q
Ans.
p → ~ q
Con: ~ q → p
Inv: ~ p → q
Cont: q → ~ p

(iv) Sum of two odd integers is even.
Ans.
If two integers are odd, then their sum is even.
Con: If the sum of two integers is even, then they are odd.
Inv: If two integers are not odd, then their sum is not even.
Cont: If the sum of two integers is not even then they are not odd.

(v) The square of an integer is a natural number.
Ans.
If a number is an integer, then its square is a natural number.
Con: If the square of a number is a natural number, then it is an integer.
Inv: If a number is not an integer, then its square is not a natural number.
Cont: If the square of a number is not a natural number then it is not an integer.

(vi) A parallelogram that is inscribed in a circle is a rectangle.
Ans.
If a parallelogram is inscribed in a circle, then it is a rectangle.
Con: If a parallelogram is a rectangle, then it is inscribed in a circle.
Inv: If a parallelogram is not inscribed in a circle then it is not a rectangle.
Cont: If a parallelogram is not a rectangle, then it is not inscribed in a circle.

(vii) The ground being wet, there has been rainfall at night.
Ans.
If the ground being wet then there has been rainfall at night.
Con: If there has been rainfall at night then the ground being wet,
Inv: If the ground is not wet then there has been no rainfall at night.
Con: If there has been no rainfall at night then the ground is not wet.

13. Are the following pair of statements negation of each other?
(i) The number π is not a rational number. The number π is not an irrational number.
(ii) The number π is a rational number. The number π is an rational number.
Solution:
(i) No (ii) Yes

14. Write the component statement of the following compound statements and check whether the compound statement is true or false.

(i) 24 is a multiple of 4 and 6

Ans.

The component statement is

p: 24 is a multiple of 4

q: 24 is a multiple of 6

The truth value of the statement is ‘True’

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

Ans.

The component statements are :

p: There is a holiday or a Sunday

q: The School is closed

The truth value of the compound statement is ‘True’.

(iii) 7 is a rational number or an irrational number.

Ans.

The component statements are

p: 7 is a rational number

q: 7 is an irrational number

The truth value of the compound statement is ‘True’.

(iv) 57 is divisible by 2 or 3.

Ans.

The component statements are

p: 57 is divisible by 2

q: 57 is divisible by 3

The truth value of the compound statement is ‘True’.

(v) All things have two eyes and two legs.

Ans.

The component statements are

p: All things have two eyes.

q: All things have two eyes

The truth value of the compound statement is ‘false’

(vi) 2 is an even number and a prime number.

Ans. The component statements are

p: 2 is an even number

q: 2 is a prime number

The truth value of the compound statement is ‘ True.

(vii) Every parallelogram is a trapezium or a rhombus.

Ans. The component statements are

p: Every parallelogram is a trapezium

q: Every parallelogram is a rhombus.

The truth value of the compound statement is “false”.

15. Identify the Quantifiers of the following statements.

(i) There exists a number that 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 of India.
(iv) For all x ∈ R , sin2 x + cos2 x = 1
(v) There exists an even prime number other than 2.
(vi) For every negative integer x, x3 is also a negative integer.
(vii) For every real number x, x2 ≠ x.
Answer.
(i) There exists
(ii) for every
(iii) There exists
(iv) for all
(v) There exists
(vi) for every
(vii) for every

Question 16. Write the negation of the following statement.

(i) Every living person is not 150 years old.

(ii) There exists x∈ N, x + 3 ≠ 10

(iii) All the students completed their homework.

(iv) There exists a number that is equal to its square.

(v) For every real number x, x + 4 is greater than x.

(vi) Everyone who lives in India is an Indian.

Answer.

(i) There exists a living person who is 150 years old.

(ii) for all x ∈ N, x + 3 ≠ 10

(iii) There exists a student who has not completed his homework.

(iv) Each number is not equal to its square.

(v) There exists a real number x, x + 4 is not greater than x.

(vi) There exists a person who lives in India but is not an Indian.

Class 11 Math Chapter 1 Mathematical Reasoning Exercise 1(b)

1. Check the validity of the following statements.

(i) p: 100 is a multiple of 5 and 4.

(ii) q: 125 is multiple of 5 and 7.

(iii) r: 60 is a multiple of 3 or 5.

Ans.

(i) Here the connective is ‘and’

Step -1: 100 is a multiple of 5 (True)

Step -2: 100 is a multiple of 4 (True)

∴ 100 is a multiple of 5 and 4 is true, i.e. the statement ‘p’ is valid.

(ii) Here the connective is ‘and’

Step – 1: 125 is a multiple of 5 (True)

Step – 2: 125 is a multiple of 7 (false)

∴ The statement ‘q’ is (false), i.e. the statement ‘q’ is not a valid statement

(iii) Here the connective is ‘or’

60 is a multiple of 3 (True)

60 is a multiple of 5 (True)

Thus the component statements are both true.

∴ The statement ‘r’: 60 is a multiple of 3 or is true, i.e. ‘r’ is a valid statement.

2. Check the validity of the statements given below by the method given against each.

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

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

(iii) ‘‘If x, y are integers such that xy is odd then both x and y are odd” (by the method of contrapositive)

(iv) “If x is an integer and x2 is even then x is also even” (method of contrapositive).

Answer.

(i) Let the given statement is false.

i.e. the sum of an irrational number and a rational number is rational.

⇒ An irrational number + a rational number = a rational number

⇒ An irrational number = A rational number + rational number = A rational number

Which is absurd.

We arrive at a contradiction.

This is due to our false assumption. Thus, the given statement is true.

(ii) Let the given statement is false.

i.e. for a natural number n > 3 n2 > 9

⇒ n2 ≤ 3 ⇒ n ≤ 3

(.’. n is a natural number which contradicts the fact that n > 3)

This contradiction is due to our false assumption.

Thus for any natural number

n > 3, n2 > 9.

(iii) Let p: x and y are integers such that xy is odd. and q: both x and y are odd. We shall prove the validity of p → q by using the method of contrapositive.

Now ~ q: It is not true that both x and y are odd, i.e. at least one of x (or y) is even.

Let ~ q is true.

Let x is even and x = 2n for some integer n.

⇒ xy = 2ny for some integer n.

⇒ xy is even (not odd)

⇒ ~ p is true.

Thus by the method of contrapositive, we proved ~ q ⇒ ~ p, i.e. p → q is true.

(iv) Let p: x is an integer and x2 is even

q : x is even.

Let ~ q is true

⇒ x is not an even integer

⇒ x is an odd integer

Let x = 2n + 1 for some integer ‘n ’

⇒ x2 = (2n + 1)2 = 4n2 + 4n + 1

= 2 (2n2 + 2n) + 1

which is odd.

⇒ p is false ⇒ ~ p is true

Thus ~ q ⇒ ~ p

∴ Thus the statement is true.

i.e x is even ⇒x is also even

3 . By giving counter-examples, show that the following statements are not true:

(i) If measures of all the angles of a triangle are equal, then the triangle is an obtuse angled triangle.

(ii) For every real number x and y, x2 = y2 ⇒ x = y.

(iii) The equation x2 – 1 = 0 does not have any root lying between 0 and 2.

Ans.

(i) Let in ΔABC

∠A = ∠B = ∠C = 60°

Clearly, no angle is obtuse.

∴ The triangle is not obtuse-angled.

(ii) for x = – 2, y = 2

x2 = y2 = 4 but x ≠ y

∴ x2 = y2 ⇒ x = y

(iii) x = 1 is a root of x2 – 1=0

that lies between 0 and 2.

∴ The given statement is false.

4. Check the validity of “If I do not work, I sleep.

If I am worried, I will not sleep. Therefore, if I am worried, I will work”.

Ans.

Method-1:

Let p: I work

q: I sleep

r: I am worried

“If I do not work, I sleep” can be written as ~ p → q

“If I am worried, I will not sleep” can be written as r → ~ q

‘If I am worried, I will work’’ can be written as r → p

The given statement can be written as [(~ p → q) ∧ (r → ~ q)] → ( r → p)

Let us draw the truth table

As [(~ p → q) ∧ (r → q)] → (r → p)

is a tautology the given statement is logically valid.

Method – 2:

Let p: I do work

q: I sleep

r: I am worried

“If I do not work, I sleep ” can be represented as ~ p → q

“If I am worried, I will not sleep” can be written as r → ~ q

Thus we can write the given statement as

[(~ p → q) ∧ (r → q)] → (r → p)

But a statement and its contrapositive are equivalent.

Thus this can be written as

[(r → ~ q) ∧ (~ q → p)] → (r → p)

By the principle of syllogism, the given statement is logically valid.

5.Let a and b be integers. By the law of contrapositive prove that if ab is even then either a is even or b is even:

Answer.

Let p: For integers ‘a and b’ ab is even q: a is even or b is even

we shall check the validity of p → q by the method of contrapositive Let ~ q is true.

⇒ Both a and b are odd

Let a = 2m + 1, b = 2n + 1

for some integers m and n

⇒ ab = (2m + 1) (2n + 1)

= 4mn + 2 (m + n ) + 1

= 2 [2mn + m + n) + 1

Which is odd

⇒ p is false

⇒ ~ p is true

Thus the method of contrapositive p → q is true.

i.e. The given statement is logically valid

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