Summary
Appendix 1 of the Class 12 Maths NCERT textbook, "Proofs in Mathematics", covers six methods of proof — three direct (straight forward, mathematical induction, proof by cases) and three indirect (contradiction, contrapositive, counterexample) — with fully worked examples for each.
- What a proof is — The appendix frames a proof as a chain of deductive steps, each justified by a definition, axiom or already-proved result. This standard of rigour is what separates a genuine mathematical argument from a plausible-sounding claim.
- Direct methods of proof — Three direct approaches start from the hypothesis: a straightforward chain of implications, mathematical induction that leaps from one case to the next along the natural numbers, and proof by cases that exhausts every possibility and proves each in turn.
- Indirect methods of proof — When a direct route is hard, the appendix reasons through an equivalent statement — assuming the opposite and deriving a contradiction, proving the contrapositive ~q ⇒ ~p, or disproving a generalisation outright with a single counterexample.
Key points & formulas
- 01A proof is a chain of deductive arguments where each statement is justified by a definition, axiom, or previously established proposition using allowed logical rules.
- 02Direct proofs have three forms: straight forward (step-by-step implication chain), mathematical induction (base case + inductive step), and proof by cases/exhaustion (split hypothesis into sub-cases r, s, t and prove each implies q).
- 03Mathematical induction rests on the axiom: if 1∈S and k∈S implies k+1∈S for a subset S of ℕ, then S = ℕ.
- 04Indirect proofs have three forms: proof by contradiction (assume negation, derive contradiction), proof by contrapositive (prove ~q⇒~p instead of p⇒q), and proof by counterexample.
- 05The contrapositive of p⇒q is ~q⇒~p, formed by interchanging hypothesis and conclusion and negating both; it is logically equivalent to the original conditional.
- 06A counterexample is a single instance that disproves a general statement; one counterexample is sufficient to invalidate a generalisation.
- 07Proof by contradiction (Reductio Ad Absurdum) assumes the given statement is false; reaching a logical contradiction shows the assumption is wrong and the original statement is therefore true.
Frequently asked questions
01What is a mathematical proof according to NCERT Class 12 Appendix 1?
A proof is a sequence of statements where each statement is justified by a definition, axiom, or a proposition previously established by deduction using allowed logical rules — forming a chain of deductive arguments with premises and conclusions.
02What are the two broad categories of proof discussed in this appendix?
Direct proof and indirect proof. A direct proof starts from what is given in the proposition; an indirect proof establishes the result by proving an equivalent proposition.
03What are the three methods of direct proof covered in this chapter?
The straight forward approach (step-by-step chain of implications from hypothesis to conclusion), mathematical induction (base case for n=1 plus inductive step from n=k to n=k+1), and proof by cases or exhaustion (splitting the hypothesis into mutually exhaustive sub-cases and proving each implies the conclusion).
04What is mathematical induction and what axiom does it rely on?
Mathematical induction is a deductive strategy for proving propositions about natural numbers. It relies on the axiom: for a subset S of ℕ, if 1∈S and k+1∈S whenever k∈S, then S = ℕ. A proof shows the base case S(1) is true, then shows S(k) true implies S(k+1) true.
05What is proof by contradiction (Reductio Ad Absurdum)?
An indirect method where you assume the given statement is false, then use logical rules to arrive at a conclusion that contradicts the assumption. The contradiction shows the assumption was wrong, proving the original statement is true. For example, assuming the set of all prime numbers is finite leads to a contradiction, so the set must be infinite.
06What is the contrapositive of a statement, and how is it used in proof?
The contrapositive of p⇒q is ~q⇒~p, formed by interchanging the hypothesis and conclusion and negating both. Since the contrapositive is logically equivalent to the original conditional, proving ~q⇒~p is sufficient to prove p⇒q.
07What is a counterexample and why is one sufficient to disprove a generalisation?
A counterexample is a specific instance that shows a general statement is false. Because disproving a proposition requires only one failure case, a single counterexample is enough. For instance, φ(x)=|x| is continuous everywhere but not differentiable at x=0, which disproves the statement 'every continuous function is differentiable'.
08What is proof by cases (proof by exhaustion)?
When the hypothesis p can be split into mutually exhaustive sub-cases r, s, t so that p = r∨s∨t, it suffices to prove r⇒q, s⇒q, and t⇒q separately. Together they cover all possibilities and so p⇒q is established.
09How does the straight forward direct approach to proof work?
Starting from the given hypothesis p, each step deduces a new statement using axioms, definitions, or previously proved theorems, forming the chain p⇒r⇒s⇒…⇒q until the conclusion q is reached.
10Is the NCERT Class 12 Maths Appendix 1 PDF free to download on this site?
Yes — the PDF is available free with no sign-up required.
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