Mathematical Reasoning : Notes and Study Materials -pdf
CBSE Class 11 Maths Notes Chapter 14 Mathematical Reasoning
Statements
A statement is a sentence which is either true or false, but not both simultaneously.
Note:
No sentence can be called a statement if
- It is an exclamation.
- It is an order or request.
- It is a question.
Simple Statements
A statement is called simple if it cannot be broken down into two or more statements.
Compound Statements
A compound statement is the one which is made up of two or more simple statement.
Connectives
The words which combine or change simple statements to form new statements or compound statements are called connectives.
Conjunction
If two simple statements p and q are connected by the word ‘and’, then the resulting compound statement “p and q” is called a conjunction of p and q is written in symbolic form as “p ∧ q”.
Note:
- The statement p ∧ q has the truth value T (true) whenever both p and q have the truth value T.
- The statement p ∧ q has the truth value F (false) whenever either p or q or both have the truth value F.
Disjunction
If two simple statements p and q are connected by the word ‘or’, then the resulting compound statement “p or q” is called disjunction of p and q and is written in symbolic form as “p ∨ q”.
Note:
- The statement p ∨ q has the truth value F whenever both p and q have the truth value F.
- The statement p ∨ q has the truth value T whenever either p or q or both have the truth value T.
Negation
An assertion that a statement fails or denial of a statement is called the negation of the statement. The negation of a statement p in symbolic form is written as “~p”.
Note:
- ~p has truth value T whenever p has truth value F.
- ~p has truth value F whenever p has truth value T.
Negation of Conjunction
The negation of a conjunction p ∧ q is the disjunction of the negation of p and the negation of q.
Equivalently we write ~ (p ∧ q) = ~p ∨ ~q.
Negation of Disjunction
The negation of a disjunction p v q is the conjunction of negation of p and the negation of q.
Equivalently, we write ~(p ∨ q) = ~p ∧ ~q.
Negation of Negation
Negation of negation of a statement is the statement itself.
Equivalently, we write ~(~p) = p
The Conditional Statement
If p and q are any two statements, then the compound statement “if p then g” formed by joining p and q by a connective ‘if-then’ is called a conditional statement or an implication and is written in symbolically p → q or p ⇒ q, here p is called hypothesis (or antecedent) and q is called conclusion (or consequent) of the conditional statement (p ⇒ q).
Contrapositive of Conditional Statement
The statement “(~q) → (~p) ” is called the contrapositive of the statement p → q.
Converse of a Conditional Statement
The conditional statement “q → p” is called the converse of the conditional statement “p → q”.
Inverse of Conditional Statement
The Conditional statement “q → p” is called inverse of p → q.
The Biconditional Statement
If two statements p and q are connected by the connective ‘if and only if’, then the resulting compound statement “p if and only if q” is called biconditional of p and q and is written in symbolic form as p ⇔ q.
Quantifier
(i) For all or for every is called universal quantifier.
(ii) There exists is called existential quantifier.
Validity of Statements
A statement is said to valid or invalid according to as it is true or false.
If p and q are two mathematical statements, then the statement
(i) “p and q” is true if both p and q are true.
(ii) “p or g” is true if p is false
⇒ q is true orq is false ⇒ p is true.
(iii) “If p, then q” is true p is true ⇒ q is true
or
q is false
⇒ p is false
or
p is true and q is false less us to a contradiction,
(iv) “p if and only if q” is true, if
(a) p is true ⇒ q is true and
(b) q is true ⇒ p is true.
Mathematical Reasoning Class 11 MCQs Questions with Answers
Question 1.
The connective in the statement 2 + 7 > 9 or 2 + 7 < 9 is
(a) and
(b) or
(c) >
(d) <
Answer
Answer: (b) or
Hint:
Given, statement is 2 + 7 > 9 or 2 + 7 < 9 Here, connective is or. It connects two statement 2 + 7 > 9, 2 + 7 < 9
Question 2.
Which of the following is not a negation of the statement A natural number is greater than zero
(a) A natural number is not greater than zero
(b) It is false that a natural number is greater than zero
(c) It is false that a natural number is not greater than zero
(d) None of these
Answer
Answer: (c) It is false that a natural number is not greater than zero
Hint:
Gievn statement is:
A natural number is greater than zero
Negation of the statement:
A natural number is not greater than zero
It is false that a natural number is greater than zero
So, option 3 is not true.
Question 3.
Which of the following is a statement
(a) x is a real number
(b) Switch of the fan
(c) 6 is a natural number
(d) Let me go
Answer
Answer: (c) 6 is a natural number
Hint:
The statement 6 is a natural number is true.
So, it is a statement.
Question 4.
The contra-positive of the statement If a triangle is not equilateral, it is not isosceles is
(a) If a triangle is not equilateral, it is not isosceles
(b) If a triangle is equilateral, it is not isosceles
(c) If a triangle is not equilateral, it is isosceles
(d) If a triangle is equilateral, it is isosceles
Answer
Answer: (d) If a triangle is equilateral, it is isosceles
Hint:
Given, statement is:
If a triangle is not equilateral, it is not isosceles.
Now, contra-positive is:
If a triangle is equilateral, it is isosceles.
Question 5.
Which of the following is a statement
(a) I will go tomorrow
(b) She will come today
(c) 3 is a prime number
(d) Tomorrow is Friday
Answer
Answer: (c) 3 is a prime number
Hint:
The statement 3 is a prime number is true.
So, it is a statement.
Question 6.
The contra-positive of the statement if p then q is
(a) if ~p then q
(b) if p then ~q
(c) if q then p
(d) if ~q then ~p
Answer
Answer: (d) if ~q then ~p
Hint:
Given statement is if p then q
Now, contra-positive of the statement is:
if ~q then ~p
Question 7.
Which of the following is not a statement
(a) The product of (-1) and 8 is 8
(b) All complex number are real number
(c) Today is windy day
(d) All of the above
Answer
Answer: (d) All of the above
Hint:
A sentence is a statement if it is true.
None of the above sentence is true.
So, option 4 is the correct answer.
Question 8.
If (p or q) is true, then
(a) p is true and q is false
(b) p is true and q is true
(c) p is false and q is true
(d) All of the above
Answer
Answer: (d) All of the above
Hint:
(p or q) is false when both p and q are false otherwise it is true.
Question 9.
Which of the following statement is a conjunction
(a) Ram and Shyam are friends
(b) Both Ram and Shyam are friends
(c) Both Ram and Shyam are enemies
(d) None of these
Answer
Answer: (d) None of these
Hint:
All the statements are conjunction. So, option 4 is the correct answer.
Question 10.
Which of the following is a compound statement
(a) Sun is a star
(b) I am a very strong boy
(c) There is something wrong in the room
(d) 7 is both odd and prime number.
Answer
Answer: (d) 7 is both odd and prime number.
Hint:
A compound statement is connected with And , or , etc.
So, the statement 7 is both odd and prime number is a compound statement.
Question 11.
Which of the following is a statement
(a) x is a real number
(b) Switch of the fan
(c) 6 is a natural number
(d) Let me go
Answer
Answer: (c) 6 is a natural number
Hint:
The statement 6 is a natural number is true.
So, it is a statement.
Question 12.
Which of the following is not a statement
(a) 8 is less than 6.
(b) Every set is finite set.
(c) The sun is a star.
(d) Mathematics is fun.
Answer
Answer: (d) Mathematics is fun.
Hint:
8 is less than 6 if false. So it is a statement.
Every set is finite set is false. So it is a statement.
The sun is a star is true. So it is a statement.
Mathematics is fun. This sentence is not always true. Hence, it is not a statement.
Question 13.
Which of the following is true
(a) A prime number is either even or odd
(b) √3 is irrational number.
(c) 24 is a multiple of 2, 4 and 8
(d) Everyone in India speaks Hindi.
Answer
Answer: (d) Everyone in India speaks Hindi.
Hint:
The statement Everyone in India speaks Hindi is not true.
This is because, there are some states like Tamilnadu, Kerala, etc. where the person does not speak Hindi.
Question 14.
If (p and q) is false then
(a) p is true and q is false
(b) p is false and q is false
(c) p is false and q is true
(d) all of the above
Answer
Answer: (d) all of the above
Hint:
(p and q) is true when both p and q are true otherwise it is false.
Question 15.
The converse of the statement p ⇒ q is
(a) p ⇒ q
(b) q ⇒ p
(c) ~p ⇒ q
(d) ~q ⇒ p
Answer
Answer: (b) q ⇒ p
Hint:
The converse of the statement p ⇒ q is
q ⇒ p
Question 16.
The negation of the statement The product of 3 and 4 is 9 is
(a) It is false that the product of 3 and 4 is 9
(b) The product of 3 and 4 is 12
(c) The product of 3 and 4 is not 12
(d) It is false that the product of 3 and 4 is not 9
Answer
Answer: (a) It is false that the product of 3 and 4 is 9
Hint:
Given, statement is The product of 3 and 4 is 9
The negation of the statement is:
It is false that the product of 3 and 4 is 9
Question 17.
Sentence involving variable time such as today, tomorrow, or yesterday are
(a) Statements
(b) Not statements
(c) may or may not be statements
(d) None of these
Answer
Answer: (b) Not statements
Hint:
Sentence involving variable time such as today, tomorrow, or yesterday are not statements. This is because it is not known what time is referred here.
Question 18.
The converse of the statement if a number is divisible by 10, then it is divisible by 5 is
(a) if a number is not divisible by 5, then it is not divisible by 10
(b) if a number is divisible by 5, then it is not divisible by 10
(c) if a number is not divisible by 5, then it is divisible by 10
(d) if a number is divisible by 5, then it is divisible by 10
Answer
Answer: (d) if a number is divisible by 5, then it is divisible by 10
Hint:
Given, statement is if a number is divisible by 10, then it is divisible by 5
Now, converse of the statement is:
if a number is divisible by 5, then it is divisible by 10
Question 19.
Which of the following is the conditional p → q
(a) q is sufficient for p
(b) p is necessary for q
(c) p only if q
(d) if q then p
Answer
Answer: (c) 6 is a natural number
Hint:
Given, p → q
Now, conditional of the statement is
p only if q
Question 20.
Which of the following is not a negation of the statement A natural number is greater than zero
(a) A natural number is not greater than zero
(b) It is false that a natural number is greater than zero
(c) It is false that a natural number is not greater than zero
(d) None of these
Answer
Answer: (c) It is false that a natural number is not greater than zero
Hint:
Given statement is:
A natural number is greater than zero
Negation of the statement:
A natural number is not greater than zero
It is false that a natural number is greater than zero
So, option 3 is not true.