CBSE Class 11 Mathematics Mathematical Reasoning Assignment Set 02

MATHEMATICAL REASONING

Type-I

Concept: Statements
A sentence which is either true or false, but not both.

Question 1. Check whether the following sentences are statements. Give reasons for your answer: (i) There are 35 days in a month. (ii) Mathematics is difficult. (iii) The sum of 5 and 7 is greater than 10. (iv) The square of a number is an even number. (v) Where is going?
Answer:
(i) It is a false sentence because a month has at most 31 days. Hence, it is a statement.
(ii) It is an opinion and varies from person to person, so it is not universally true or false. Hence, it is not a statement.
(iii) The sum of 5 and 7 is 12, which is greater than 10. This is always true. Hence, it is a statement.
(iv) The square of a number can be even or odd (e.g., \(2^2 = 4\) is even, but \(3^2 = 9\) is odd). Since it is not definitively true or false for all numbers, it is not a statement.
(v) It is an interrogative sentence. Hence, it is not a statement.
In simple words: A sentence is a statement only if it can be clearly categorized as definitively true or definitively false.

Exam Tip: Exclamatory, interrogative, imperative sentences, and sentences involving variable opinions or pronouns (like "he", "today") without context are never mathematical statements.

Type-II

Negation of a Statement
Denial of a statement is called negation of a statement.

Question 1. Write the negations of the following statements: (i) Chennai is the capital of Tamil Nadu. (ii) \(\sqrt{2}\) is not a complex number. (iii) All triangles are equilateral triangles. (iv) The number 2 is greater than 7. (v) Every natural number is an integer.
Answer:
(i) Chennai is not the capital of Tamil Nadu.
(ii) \(\sqrt{2}\) is a complex number.
(iii) Not all triangles are equilateral triangles (or, There exists a triangle which is not an equilateral triangle).
(iv) The number 2 is not greater than 7 (or, \(2 \le 7\)).
(v) Not every natural number is an integer (or, There exists a natural number which is not an integer).
In simple words: To negate a statement, you insert "not" or change words like "all" to "not all" or "there exists".

Exam Tip: When negating universal quantifiers ("all", "every"), use existential phrasing ("there exists at least one") rather than jumping straight to "none".

Type-III

Compound Statement and Component Statement
A compound statement is a statement which is made up of two or more statements. Each statement is called a component statement.

Question 1. Identify the component statements and the connective in the following compound statement: "Number 3 is prime and odd."
Answer:
Component statement 1 (\(p\)): Number 3 is prime.
Component statement 2 (\(q\)): Number 3 is odd.
Connective: "and"
In simple words: Break the long sentence into simpler independent facts and find the connecting word.

Exam Tip: Always state the individual component sentences so that each can stand alone as a complete mathematical statement.

Type-IV

Compound Statement with 'and' or 'or'

Question 1. Find the component statements of the following compound statement and check whether they are true or false: "All rational numbers are real and all real numbers are complex."
Answer:
Component statement 1 (\(p\)): All rational numbers are real. (True)
Component statement 2 (\(q\)): All real numbers are complex. (True)
Since both component statements are true, the given compound statement is true.
In simple words: A compound statement connected by "and" is true only when all its parts are true.

Exam Tip: For an "and" statement to be true, every single component must be true. If even one component is false, the entire compound statement is false.

Type-V

Quantifiers
Quantifiers are phrases like 'there exists' and 'for all' etc.

Question 1. Rewrite each of the following statements with "there exists" and "for all" in appropriate places: (i) For every real number \(x\), \(x^2 \ge 0\). (ii) There is a number \(x\) such that \(x + 3 = 8\).
Answer:
(i) For all real numbers \(x\), we have \(x^2 \ge 0\).
(ii) There exists a number \(x\) such that \(x + 3 = 8\).
In simple words: Match standard mathematical phrases with their formal quantifier terminology.

Exam Tip: Universal quantifier symbol is \(\forall\) ("for all") and existential quantifier symbol is \(\exists\) ("there exists").

Type-VI

Inclusive or Exclusive Or

Question 1. State whether the 'or' used in the following statement is inclusive or exclusive: "To enter a country, you need a passport or a voter registration card."
Answer:
The 'or' used here is inclusive. A person can enter the country if they have a passport, if they have a voter registration card, or if they possess both.
In simple words: Inclusive means "one or the other or both", while exclusive means "one or the other, but not both".

Exam Tip: In mathematics, "or" is almost always interpreted as inclusive unless explicitly stated otherwise.

Type-VII

Implications

Question 1. Rewrite the following statement with "if-then" in five different ways: "If a number is a multiple of 9, then it is a multiple of 3."
Answer:
Let \(p\): A number is a multiple of 9.
Let \(q\): It is a multiple of 3.
The five different ways to express the implication \(p \implies q\) are:
1. If a number is a multiple of 9, then it is a multiple of 3.
2. A number is a multiple of 9 only if it is a multiple of 3.
3. A number being a multiple of 9 is a sufficient condition for it to be a multiple of 3.
4. A number being a multiple of 3 is a necessary condition for it to be a multiple of 9.
5. Ensuring a number is not a multiple of 3 implies that it is not a multiple of 9 (\(\sim q \implies \sim p\)).
In simple words: An implication can be phrased using 'if-then', 'only if', 'sufficient', or 'necessary'.

Exam Tip: Memorize the equivalence between \(p \implies q\), "p only if q", "p is sufficient for q", and "q is necessary for p".

Type-VIII

Contrapositive and Converse Statement
Contrapositive statement of \(p \implies q\) is \(\sim q \implies \sim p\).
Converse of the statement \(p \implies q\) is \(q \implies p\).

Question 1. Write the converse and contrapositive of the statement: "If a number \(n\) is even, then \(n^2\) is even."
Answer:
Given statement (\(p \implies q\)): If a number \(n\) is even, then \(n^2\) is even.
Converse (\(q \implies p\)): If \(n^2\) is even, then the number \(n\) is even.
Contrapositive (\(\sim q \implies \sim p\)): If \(n^2\) is not even, then the number \(n\) is not even.
In simple words: Converse flips the two parts around. Contrapositive flips them and negates both.

Exam Tip: A conditional statement and its contrapositive are logically equivalent, but its converse is not necessarily equivalent to the original statement.

Type-IX

Validating Statements
Direct method: By assuming that \(p\) is true, prove that \(q\) must be true.
Contrapositive method: By assuming \(q\) is false, prove that \(p\) must be false.
Method of contradiction: Assume \(p\) is not true. Then we arrive at some result which contradicts our assumption.

Question 1. Show that the statement "If \(x\) is a real number such that \(x^3 + 4x = 0\), then \(x = 0\)" is validated by the direct method.
Answer:
Assume that \(p\) is true, i.e., \(x\) is a real number such that \(x^3 + 4x = 0\).
We can factor the equation as:
\(x(x^2 + 4) = 0\)
Since \(x\) is a real number, \(x^2 \ge 0\), which means \(x^2 + 4 \ge 4\) (so \(x^2 + 4 \neq 0\)).
Therefore, we must have:
\(x = 0\)
Thus, \(q\) is true. The statement is validated directly.
In simple words: Start with the given assumption and directly solve or deduce the final conclusion.

Exam Tip: In the direct method, clearly state your starting hypothesis and show logical algebraic steps leading directly to the final target equation.

Type-X

Counterexample
By giving a counterexample we can disprove a given statement.

Question 1. Disprove the following statement by giving a counterexample: "All prime numbers are odd numbers."
Answer:
Counterexample: The number 2 is a prime number because its only divisors are 1 and 2. However, 2 is an even number, not an odd number.
Hence, the statement "All prime numbers are odd numbers" is false.
In simple words: Find just one single case where the rule fails, and the statement is successfully disproved.

Exam Tip: When asked to disprove a universal statement, look for edge cases such as 0, 1, or 2 in number theory problems.

Type-XI

Validating Compound Statements

Question 1. Validate the statement by method of contradiction: "The sum of an irrational number and a rational number is irrational."
Answer:
Let \(x\) be a rational number and \(y\) be an irrational number. We want to show that \(x + y\) is irrational.
Assume to the contrary that \(x + y\) is a rational number, say \(r\).
Then:
\(x + y = r\)
Since \(x\) and \(r\) are rational numbers, we can express \(y\) as:
\(y = r - x\)
The difference of two rational numbers is always rational. Hence, \(r - x\) is rational.
This implies that \(y\) is a rational number, which contradicts our given hypothesis that \(y\) is an irrational number.
Therefore, our assumption that \(x + y\) is rational is false.
Thus, the sum of a rational number and an irrational number must be irrational.
In simple words: Assume the opposite of what you want to prove, show that it leads to a contradiction, and conclude that your original statement must be correct.

Exam Tip: Always begin a proof by contradiction with the phrase "Assume to the contrary that..." followed by the negation of the statement to be proved.