Consider the statement, “Either – 2 ≤ x ≤ – 1 or 1 ≤ x ≤2”. The negation of this statement is
x < – 2 or 2 < x or – 1 < x < 1
The statement is a disjunction P ∨ Q with P: \(-2 \le x \le -1\) and Q: \(1 \le x \le 2\). By De Morgan's law its negation is \(\neg P \wedge \neg Q\).
Step 1 — \(\neg P\): \(x \lt -2\) or \(x \gt -1\).
Step 2 — \(\neg Q\): \(x \lt 1\) or \(x \gt 2\).
Step 3 — Intersect the two: x must lie outside both closed intervals [–2, –1] and [1, 2]. On the real line that leaves \(x \lt -2\), \(-1 \lt x \lt 1\) and \(x \gt 2\), i.e. \(x \lt -2 \text{ or } 2 \lt x \text{ or } -1 \lt x \lt 1\).
Leaving out the middle gap \(-1 \lt x \lt 1\) misses values such as x = 0, which also make the statement false. Keeping only the middle gap misses x = 5 and x = –5. Using \(x \le -2\) and \(2 \le x\) wrongly includes x = –2 and x = 2, where the original statement is true.
x < – 2 or 2 < x or – 1 < x < 1 option is correct.
Similar Questions
The resolvent of the set of clauses
\((A \lor B,\ \sim A \lor D,\ C \lor \sim B)\) is
In Propositional Logic, given P and \(P \to Q\), we can infer __________.
A horn clause is __________.
Which of the following arguments are not valid ?
(a) “ If Gora gets the job and works hard, then he will be promoted. If Gora gets promotion, then he will be happy. He will not be happy, therefore, either he will not get the job or he will not work hard”.
(b) “Either Puneet is not guilty or Pankaj is telling the truth. Pankaj is not telling the truth, therefore, Puneet is not guilty”.
(c) If n is a real number such that \(n \gt 1\), then \(n^{2} \gt 1\). Suppose that \(n^{2} \gt 1\), then \(n \gt 1\).
Codes :
The clausal form of the disjunctive normal form \(\neg A \vee \neg B \vee \neg C \vee D\) is :
“If my computations are correct and I pay the electric bill, then I will run out of money. If I don’t pay the electric bill, the power will be turned off. Therefore, if I don’t run out of money and the power is still on, then my computations are incorrect.”
Convert this argument into logical notations using the variables c, b, r, p for propositions of computations, electric bills, out of money and the power respectively. (Where ¬ means NOT)
Give a compound proposition involving propositions p, q and r that is true when exactly two of p, q and r are true and is false otherwise.
How many different truth tables of the compound propositions are there that involve the propositions p & q ?
Let P be “It is hot day” and q be “The temperature is 48°C”. Write in simple sentences the meaning of ~p∧~q.
In propositional logic if (P → Q) ∧ (R → S) and (P ∨ R) are two premises such that
\(\dfrac{\begin{array}{c}(P → Q) ∧ (R → S) \\ P ∨ R\end{array}}{Y}\)
Y is the premise :
Important Questions from Propositional Logic
- Assume that a Creative ($C$) person will Succeed ($S$) if the person is also Disciplined ($D$), but will not succeed otherwise. Now, consider the following statements:
(i) $C \land S \Leftrightarrow D$
(ii) $C \Rightarrow (S \Leftrightarrow D)$
(iii) $C \Leftrightarrow ((D \Rightarrow S) \lor \neg S)$
Which of the following options is correct? - Let $x$ and $y$ be two propositions. Which of the following statements is a tautology /are tautologies?
Let $p$ and $q$ be any two propositions. Consider the following propositional statements.
$S_1: p \to q$ $S_2: \neg p \land q$ $S_3: \neg p \lor q$ $S_4 : \neg p \lor \neg q$
where $\land$ denotes conjunction (AND operation), $\lor$ denotes disjunction (OR operation), and $\neg$ denotes negation (NOT operation). Which one of the following options is correct?
(Note: $\equiv$ denotes logical equivalence)Which of the following logical implications are tautology?
A. $[\sim p \land (p \lor q)] \rightarrow q$
B. $\sim p \rightarrow (p \rightarrow q)$
C. $[p \land (p \rightarrow q)] \rightarrow q$
D. $\sim (p \rightarrow q) \rightarrow q$
Choose the correct answer from the options given below:
- Sentence $X$ is said to entail Sentence $Y$ if whenever $X$ is $TRUE$, $Y$ also must hold $TRUE$.
Which of the following statements is/are correct if $X$ entails $Y$?
