Match the following terms : Codes :List - I List - II (a) Vacuous proof (i) A proof that the implication \(p \rightarrow q\) is true based on the fact that p is false. (b) Trivial proof (ii) A proof that the implication \(p \rightarrow q\) is true based on the fact that q is true. (c) Direct proof (iii) A proof that the implication \(p \rightarrow q\) is true that proceeds by showing that q must be true when p is true. (d) Indirect proof (iv) A proof that the implication \(p \rightarrow q\) is true that proceeds by showing that p must be false when q is false.
| a | b | c | d |
| (i) | (ii) | (iii) | (iv) |
A vacuous proof uses a false hypothesis p, since \(p \rightarrow q\) is then automatically true; so (a) matches (i).
A trivial proof uses a true conclusion q, which makes the implication true regardless of p; so (b) matches (ii).
A direct proof assumes p and derives q, so (c) matches (iii); an indirect proof (contraposition) assumes not q and derives not p, so (d) matches (iv).
Hence, (a)-(i), (b)-(ii), (c)-(iii), (d)-(iv).
Similar Questions
Which one of the following is true ?
Consider a proposition given as :
“ \(x \geq 6\), if \(x^{2} \geq 25\) and its proof as :
If \(x \geq 6\), then \(x^{2} = x \cdot x \geq 6 \cdot 6 = 36 \geq 25\)
Which of the following is correct w.r.to the given proposition and its proof ?
(a) The proof shows the converse of what is to be proved.
(b) The proof starts by assuming what is to be shown.
(c) The proof is correct and there is nothing wrong.
The mathematical notation to describe logical entailment of a scentence “α entails another sentence β” is
Match the following :
List – I List – II a. Absurd i. Clearly impossible being contrary to some evident truth. b. Ambiguous ii. Capable of more than one interpretation or meaning. c. Axiom iii. An assertion that is accepted and used without a proof. d. Conjecture iv. An opinion preferably based on some experience or wisdom. Codes :
Important Questions from Mathematical Logic
Which of the following equation represents a function?
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- \(\sqrt[3]{{\sqrt {0.000729} }} =\)
