ilovepdf_merged (6).pdf). Content covers sections 38.1–38.x.Mathematical reasoning studies statements (sentences that are true or false, not both), how they combine, and how to prove or disprove them.
Commands, questions, and exclamations are not statements. Negation of p is ~p (“not p”). Double negation returns p.
p ⇒ q (“if p then q”): false only when p true and q false. Equivalent readings: p is sufficient for q; q is necessary for p.
Direct proof, contrapositive proof, contradiction, and counter-example for disproof. Quantifiers: “for every” vs “there exists” reverse under negation.
Most exam-important points from this chapter:
Start every answer with the key definition or standard form from Mathematical Reasoning.
Copy the formula, then substitute values; never jump to the number alone.
Check domain, quadrant, non-zero denominators, and applicability of the theorem.
Memorise and apply: already discussed the inductive reasoning in mathematical induction. Now, we shall discuss
Memorise and apply: Every square is a rectangle.
English questions from NIOS Mathematics (311) public / sample papers (Sample 2024, Apr 2024, Oct 2024, Apr 2025; board-style where scanned papers had no extractable text). Mapped exclusively to this chapter — no cross-chapter duplicates. Use Model Answer for the marking key and Explanation for working.
6 question(s) · Sources: Board-style counter-example, Board-style implication truth, Board-style negation, Board-style statement, Sample QP 2024
PYQ1. Converse of “If n is an odd number, then n² is also odd” is:
Model Answer
1) If n² is an odd number, then n is also an odd number
Explanation
1) Converse of p⇒q is q⇒p.
PYQ2. Contrapositive of “If a number is divisible by 6, then it is divisible by 3” is:
Model Answer
1) If a number is not divisible by 3, then it is not divisible by 6
Explanation
1) Contrapositive of p⇒q is ~q⇒~p.
PYQ3. Which of the following is a mathematical statement? (i) Close the door (ii) 2+2=4 (iii) Where are you?
Model Answer
1) (ii) only
Explanation
1) Only (ii) has a definite truth value (True). Commands and questions are not statements.
PYQ4. Write the negation of: “All natural numbers are even.”
Model Answer
1) There exists a natural number that is not even (some natural number is odd).
Explanation
1) Negation of ∀ is ∃ with negated predicate — not “all are odd”.
PYQ5. If p is true and q is false, what is the truth value of p ⇒ q?
Model Answer
1) False
Explanation
1) Implication is false only when antecedent is true and consequent is false.
PYQ6. Disprove: “Every prime number is odd.”
Model Answer
1) Counter-example: 2 is prime and even
Explanation
1) A single counter-example refutes a universal claim.
10 English exam-style problems for this chapter (NIOS Mathematics 311). Each question states the formula, gives full step-by-step working with the final answer, plus formal/easy explanations. English only.
Which of the following is a statement? (i) Close the door (ii) 2+2=4 (iii) Where are you?
1) (i) command — not a statement.
2) (iii) question — not a statement.
3) (ii) has truth value True — is a statement.
Answer: (ii) 2+2=4 is a statement
Statements are declarative sentences that are true or false.
Only the equation has a clear T/F value.
Imperatives and questions are excluded.
Open sentences with free variables need quantifiers to become statements.
Write the negation of: “All natural numbers are even.”
1) Negation: “There exists a natural number that is not even” (i.e. some natural number is odd).
Answer: ∃ a natural number that is not even
Negation of ∀ is ∃ with negated predicate.
Not all even means at least one odd.
Do not write “all are odd” (stronger, not equivalent).
Quantifier flip is the key.
p: 15 is multiple of 5 (T). q: 15 is multiple of 4 (F). Is p∧q true?
1) p true, q false ⇒ p∧q false.
Answer: False
Conjunction requires both components true.
One false kills AND.
p∨q would be true here.
Check each part carefully.
p true, q false. Truth value of p⇒q?
1) Implication is false only in the true→false case ⇒ false.
Answer: False
Truth table of implication.
True cannot force a false conclusion.
If p false, p⇒q is true regardless of q.
Do not confuse with converse.
Write the contrapositive of: If a number is divisible by 9, then it is divisible by 3.
1) If a number is not divisible by 3, then it is not divisible by 9.
Answer: If not divisible by 3, then not divisible by 9
Contrapositive negates and swaps.
Flip and negate both sides.
Logically equivalent to the original implication.
Converse would be “if divisible by 3 then by 9” (false in general).
Disprove: “Every prime number is odd.”
1) Counter-example: 2 is prime and even.
2) Hence the universal claim is false.
Answer: False; counter-example n=2
A single counter-example refutes a universal statement.
Two is the even prime.
Existence claims need different methods.
Must verify the example really is prime and even.
Is “x + 2 = 5” a statement if x is an unspecified real number?
1) It is an open sentence; truth depends on x.
2) Not a statement until x is fixed or quantified.
Answer: No — open sentence (not a statement as given)
Statements must be true or false definitively.
“There exists x…” would be a statement.
Free variables block truth value.
Check each algebraic step carefully.
Negate: “The number is even and positive.”
1) Not even or not positive (De Morgan).
Answer: The number is not even, or it is not positive (or both)
Negation of a conjunction is a disjunction of negations.
Break the “and”.
Do not only negate one part.
Check each algebraic step carefully.
Write the converse of: If a figure is a square, then it is a rectangle.
1) If a figure is a rectangle, then it is a square.
Answer: If rectangle, then square
Converse swaps hypothesis and conclusion.
Not equivalent to the original (false in general).
Contrapositive would be “if not rectangle then not square”.
Check each algebraic step carefully.
p false, q true. Truth value of p ∨ q?
1) OR is true if at least one is true ⇒ true.
Answer: True
Disjunction truth table.
One true is enough.
AND would be false here.
Check each algebraic step carefully.