The mathematical notation to describe logical entailment of a scentence “α entails another sentence β” is
α ⊨ β
Entailment is the semantic consequence relation: a sentence α entails a sentence β when β is true in every model (interpretation) in which α is true. In logic and in knowledge-based agents (Russell and Norvig) this is written with the double turnstile symbol ⊨, read 'α entails β' or 'β follows logically from α'.
Step 1 — The direction matters: α ⊨ β means M(α) ⊆ M(β), i.e. every model of α is also a model of β. The sentence that does the entailing is written on the left.
Step 2 — β ⊨ α states the reverse relation (β entails α), which is a different claim and is not implied by α ⊨ β.
Step 3 — α ⊆ β and β ⊆ α are subset relations between sets. They can describe the model sets M(α) and M(β), but they are not the notation for a sentence entailing another sentence.
α ⊨ β option is correct.
Similar Questions
Match the following terms :
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. Codes :
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.
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?
The value of \({4^{2{{\log }}3}}\) is
If \(\sqrt {1 + \frac{x}{{225}}} = \frac{{17}}{{15}}\) , then the value of x is
If \(x = 7 - 4\sqrt 3 \) then the value of \(x + \frac{1}{x}\) is
- \(\sqrt[3]{{\sqrt {0.000729} }} =\)
