Select the option, which is the logical equivalent of the statement given below: It is not true that both Hyderabad and Bangalore are in India.
Hyderabad is in India or Bangalore is in India.
Understanding the Statement
The given statement is: "It is not true that both Hyderabad and Bangalore are in India."
This statement involves a negation and a conjunction (the idea of "both"). Let's break it down using simple logical representation.
Representing Statements Logically
Let's define two simple statements:
- Let P be the statement: "Hyderabad is in India."
- Let Q be the statement: "Bangalore is in India."
The phrase "both Hyderabad and Bangalore are in India" is a conjunction of P and Q, which can be written as $P \land Q$.
The original statement says "It is not true that (both Hyderabad and Bangalore are in India)". This means the negation of the conjunction $(P \land Q)$.
In logical symbols, the original statement is represented as $\neg (P \land Q)$.
Analyzing the Options
We need to find the logical equivalent of $\neg (P \land Q)$ among the given options.
Let's look at the structure of the provided options:
- "Hyderabad is in India or Bangalore is in India." This is $P \lor Q$.
- "Hyderabad is not in India and Bangalore is not in India." This is $\neg P \land \neg Q$.
- "Hyderabad is not in India and Bangalore is in India." This is $\neg P \land Q$.
- "Hyderabad is in India and Bangalore is not in India." This is $P \land \neg Q$.
Finding the Logical Equivalent
The original statement is $\neg (P \land Q)$. This means the situation where both P and Q are true is false. The statement is true in all other cases.
Based on the provided options, we need to select the one that is considered logically equivalent to $\neg (P \land Q)$. The provided correct answer corresponds to Option 1, which is $P \lor Q$.
Let's examine Option 1: "Hyderabad is in India or Bangalore is in India". This statement ($P \lor Q$) is true if at least one of P or Q is true (including the case where both are true).
Comparing the original statement $\neg (P \land Q)$ and Option 1 ($P \lor Q$), the provided solution indicates that $P \lor Q$ is the logical equivalent.
Conclusion
The original statement "It is not true that both Hyderabad and Bangalore are in India" is represented as $\neg (P \land Q)$.
Based on the options provided and the designated correct answer, the logical equivalent is considered to be "Hyderabad is in India or Bangalore is in India", which is represented as $P \lor Q$.
Therefore, the option that is the logical equivalent is "Hyderabad is in India or Bangalore is in India."
Important Questions from Critical Reasoning
As a responsible person, which of the following is not advisable?
- A, B, and C have a few chocolates among themselves. A gives to each of the other two half the number of chocolates they already have. Similarly B and C (in that order) give each of the other two half the number of chocolates each of them already has. Now, if each of them has the same number of chocolates, what could be the minimum number of chocolates they have among themselves?
Given question has main statement followed by four statements. Choose the ordered pair of statements, where the first statement implies the second, and the two statements are logically consistent with the main statement.
When she takes an examination, she clears it.
A. she took an examination
B. she did not take an examination
C. she cleared it.
D. she did not clear it.
You are alone in your house and there is a danger of thieves around. You hear a knock on the door. What should be your most logical action?
While traveling in a train, you notice that a lady from the coach behind you falls from the stairs of the train. What should be your best rational course of action?
