Mathematics · Class 11 Science
Ch 14Mathematical Reasoning — Class 11 Mathematics, concept-first.
Reasoning is what separates a mathematical argument from a mere guess. Every proof, every valid conclusion in mathematics rests on the ability to move logically from what is known to what must follow. This chapter builds the vocabulary and the toolkit for that ability.
Key concepts
Hover a concept to preview it and jump to its most relevant Q&A.
Statements
A sentence qualifies as a mathematical statement only if it is either true or false, but never both and never neither.
Most relevant Q&A
- Check whether the following sentences are statements. Give reasons for your answer. (i) 8 is less than 6. (ii) Every set is a finite set. (i…Preview
- Which of the following sentences are statements? Give reasons for your answer. (i) There are 35 days in a month. (ii) Mathematics is difficu…Free
- Give three examples of sentences which are not statements. Give reasons for the answers.Free
- 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 circ…Preview
Chapter contents
The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.
Introduction
Reasoning is what separates a mathematical argument from a mere guess. Every proof, every valid conclusion in mathematics rests on the ability to move logically from what is known to what must follow.…
Statements
The raw material of mathematical reasoning is the statement. Not every sentence in English qualifies as one, so the first job is to pin down exactly what does.
New Statements from Old
Once we can recognise a statement, the next step is learning how to manufacture new statements out of the ones we already have.
Negation of a statement
The negation of a statement is simply its denial — the statement that says the opposite of what the original says.
+−Worked Examplesi2 questions
- Example 2Write the negation of the following statements. (i) Both the diagonals of a rectangle have the same length. (ii) $\sqrt{7}$ is rational.Free
- Example 3Write the negation of the following statements and check whether the resulting statements are true. (i) Australia is a continent. (ii) There…Preview
Compound statements
Many statements we meet are not simple, single ideas but are built by joining two or more smaller statements together with connecting words such as "and" and "or".
+−Worked Examplesi2 questions
- Example 4Find the component statements of the following compound statements. (i) The sky is blue and the grass is green. (ii) It is raining and it is…Free
- Example 5Find the component statements of the following and check whether they are true or not. (i) A square is a quadrilateral and its four sides eq…Preview
Special Words/Phrases
Certain words recur constantly inside compound statements — most importantly "And" and "Or" — and mathematics gives each of them one fixed, unambiguous meaning, called a connective.
The word “And”
The connective "And" joins two or more component statements into one, and it has a simple, strict rule for truth:
The word “Or”
English uses the word "Or" in two genuinely different senses, and mathematics needs to distinguish them before any compound statement built from "or" can be judged true or false.
+−Worked Examplesi2 questions
- Example 7For each of the following statements, determine whether an inclusive “Or” or exclusive “Or” is used. Give reasons for your answer. (i) To en…Free
- Example 8Identify the type of “Or” used in the following statements and check whether the statements are true or false: (i) $\sqrt{2}$ is a rational…Preview
Quantifiers
Beyond "And" and "Or", mathematical statements frequently use two special phrases called quantifiers: "there exists" and "for every" (also written "for all").
Implications
A very large share of mathematical statements take the form "if p, then q", and understanding exactly what this claims — and what it does not claim — is essential.
Contrapositive and converse
Two further statements can be built out of any "if p, then q" statement: its contrapositive and its converse. They are easy to confuse but behave very differently.
+−Worked Examplesi4 questions
- Example 9Write the contrapositive of the following statement: (i) If a number is divisible by 9, then it is divisible by 3. (ii) If you are born in I…Free
- Example 10Write the converse of the following statements. (i) If a number n is even, then n$^2$ is even. (ii) If you do all the exercises in the book,…Free
- Example 11For each of the following compound statements, first identify the corresponding component statements. Then check whether the statements are…Preview
- Example 12Given below are two pairs of statements. Combine these two statements using “if and only if”. (i) p: If a rectangle is a square, then all it…Preview
Validating Statements
Recognising a statement is only half the task; the other half is deciding whether it is true. Doing this carefully depends on identifying exactly which special words appear in the statement — "and"/"o…
+−Worked Examplesi2 questions
By Contradiction
Beyond the direct, contrapositive and "and/or" checks of the previous section, there are two further techniques for validating (or invalidating) a statement.
Miscellaneous Examples
This closing set of examples does not introduce any new idea — instead, it draws together everything built up across the chapter: recognising statements, forming negations, breaking compound statement…
+−Examples 14.Misci4 questions
- Example 17Check whether “Or” used in the following compound statement is exclusive or inclusive? Write the component statements of the compound statem…Free
- Example 18Write the negation of the following statements: (i) p: For every real number x, x$^2$ > x. (ii) q: There exists a rational number x such tha…Free
- Example 19Using the words “necessary and sufficient” rewrite the statement “The integer n is odd if and only if n$^2$ is odd”. Also check whether the…Preview
- Example 20For the given statements identify the necessary and sufficient conditions. t: If you drive over 80 km per hour, then you will get a fine.Preview
Exercises
+−Show 25 questionsHide questions25 questions
- Q1Which of the following sentences are statements? Give reasons for your answer. (i) There are 35 days in a month. (ii) Mathematics is difficu…Free
- Q2Give three examples of sentences which are not statements. Give reasons for the answers.Free
- Q3Write the negation of the following statements: (i) Chennai is the capital of Tamil Nadu. (ii) $\sqrt{2}$ is not a complex number. (iii) All…Free
- Q4Are 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…Preview
- Q5Find the component statements of the following compound statements and check whether they are true or false. (i) Number 3 is prime or it is…Preview
- Q6For each of the following compound statements first identify the connecting words and then break it into component statements. (i) All ratio…Preview
- Q7Identify the quantifier in the following statements and write the negation of the statements. (i) There exists a number which is equal to it…Preview
- Q8Check 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…Preview
- Q9State whether the “Or” used in the following statements is “exclusive” or “inclusive”. Give reasons for your answer. (i) Sun rises or Moon s…Preview
- Q10Rewrite the following statement with “if-then” in five different ways conveying the same meaning. If a natural number is odd, then its squar…Preview
- Q11Write the contrapositive and converse of the following statements. (i) If x is a prime number, then x is odd. (ii) If the two lines are para…Preview
- Q12Write each of the following statements in the form “if-then”: (i) You get a job implies that your credentials are good. (ii) The Bannana tre…Preview
- Q13Given statements in (a) and (b). Identify the statements given below as contrapositive or converse of each other. (a) If you live in Delhi,…Preview
- Q14Show that the statement p: “If x is a real number such that x$^3$ + 4x = 0, then x is 0” is true by (i) direct method, (ii) method of contra…Preview
- Q15Show that the statement “For any real numbers a and b, a$^2$ = b$^2$ implies that a = b” is not true by giving a counter-example.Preview
- Q16Show that the following statement is true by the method of contrapositive. p: If x is an integer and x$^2$ is even, then x is also even.Preview
- Q17By giving a counter example, show that the following statements are not true. (i) p: If all the angles of a triangle are equal, then the tri…Preview
- Q18Which 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 circ…Preview
- Q19Write the negation of the following statements: (i) p: For every positive real number x, the number x – 1 is also positive. (ii) q: All cats…Preview
- Q20State the converse and contrapositive of each of the following statements: (i) p: A positive integer is prime only if it has no divisors oth…Preview
- Q21Write each of the statements in the form “if p, then q”: (i) p: It is necessary to have a password to log on to the server. (ii) q: There is…Preview
- Q22Rewrite 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 you…Preview
- Q23Given below are two statements p : 25 is a multiple of 5. q : 25 is a multiple of 8. Write the compound statements connecting these two stat…Preview
- Q24Check the validity of the statements given below by the method given against it. (i) p: The sum of an irrational number and a rational numbe…Preview
- Q25Write the following statement in five different ways, conveying the same meaning. p: If a triangle is equiangular, then it is an obtuse angl…Preview