Correct Answer - Option 3 : proof by contradiction
Mathematics is the study of numbers, shapes, quantities, and patterns. Mathematics is the ‘queen of all sciences’ and its presence is there in all the subjects. It acts as the basis and structure of other subjects.
Proof of a Mathematical Statement consists of a sequence of statements, each statement being justified with a definition or an axiom or a proposition that is previously established by the method of deduction using only the allowed logical rules. Thus, each proof is a chain of deductive arguments each of which has its premises and conclusions.
Proof by contradiction:
- This method is called proof by contradiction. It is also called reductio ad absurdum (a Latin phrase) because it relies on reducing a given assumption to an absurdity.
- Here, we start with the assumption that the given statement is false. By rules of logic, we arrive at a conclusion contradicting the assumption, and hence it is inferred that the assumption is wrong and hence the given statement is true.
- In this method, to prove q is true, we start by assuming that q is false (i.e., ~ q is true). Then, by a logical argument we arrive at a situation where a statement is true as well as false, i.e., we reach a contradiction r ∧ ~ r for some statement that is always false. This can only happen when ~ q is false also. Therefore, q must be true.
- Prove that there is no rational number whose square is 2. For this, we begin by assuming that q is a rational number whose square is 2 and ended up with the conclusion that the square of q (rational number) is not 2.
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Proof by induction:- In this method of proof, examples, and steps are taken to prove a statement without the help of any theorem. In this method, the proof is made by way of induction.
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Direct proof: In this method, a given statement or equation can be proved directly from examples. By using hypothetical values, figures or data, the direct proof method can be used to prove the Sum of two even integers is always even. Example: 2 + 4 = 6. Here, we have taken 2 even numbers 2 and 4 and get result 6 which is again an even number.
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Counter positive proof:- In this method of proof, inferences are used in proofs for some given conditional statements. The contrapositive of a conditional can be formed by interchanging the conclusion and the hypothesis and then negating both of them.
Thus from the above-mentioned points, it is clear that this type of proof is proof by contradiction.
