Problem 7 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Negation of "Paris is in France and London is in England" is (A) Paris is in England and London is in France. (B) Paris is not in France or London is not in England.

Step-by-Step Solution

Answer

The correct negation is (B) "Paris is not in France or London is not in England."

Step 1: Understanding the Statement

The original statement is "Paris is in France and London is in England". This is a conjunction of two propositions: Proposition 1: "Paris is in France" and Proposition 2: "London is in England". We need to negate this entire conjunctive statement.

Step 2: Applying De Morgan's Laws

To negate a conjunction ("and" statement), we use De Morgan's Laws, which state that the negation of a conjunction is the disjunction of the negations. Thus, the negation of a statement "A and B" is "not A or not B".

Step 3: Negating Each Proposition

Apply the rule: `Negation of (Proposition 1 and Proposition 2)` becomes `Negation of Proposition 1 or Negation of Proposition 2`. Thus, "Paris is not in France or London is not in England."

Step 4: Choosing the Correct Option

Compare the negated statement with the provided options. Option (B) states "Paris is not in France or London is not in England," which matches our negated statement fully.

Key Concepts

De Morgan's Laws

De Morgan's Laws are fundamental rules in logic and set theory that describe how the negation of logical conjunctions and disjunctions must be handled. Named after Augustus De Morgan, these laws are crucial for transforming logical statements into their opposites. Here's how they work for the conjunction and disjunction: - **Negation of a conjunction**: If you have a statement "A and B," the negation would be "not A or not B." - **Negation of a disjunction**: Conversely, if you have a statement "A or B," the negation would be "not A and not B." These laws are particularly useful when working with logical propositions in mathematics and computer science. In the context of our exercise, De Morgan's Laws help us transform the original statement, "Paris is in France and London is in England," into its negation: "Paris is not in France or London is not in England." This process shows the power of these laws to provide clarity and correctness in negating logical expressions.

Logical Conjunction

Logical conjunction is the operation of combining two propositions with the word "and." It results in a new proposition that is only true if both original propositions are true. In symbolic form, a conjunction is often represented as "A ∧ B." The truth table for a conjunction is: - If both A and B are true, then the statement "A and B" is true. - If either A or B is false, or if both are false, then the statement "A and B" is false. In our exercise example, the statements "Paris is in France" and "London is in England" are joined by a conjunction to form a logical expression that asserts both are true simultaneously. Understanding conjunctions is essential to grasp how propositions interact and are evaluated in logical reasoning.

Logical Disjunction

Logical disjunction refers to the operation of combining two propositions with the word "or." Unlike conjunctions, a disjunction is true if at least one of the individual propositions is true. Symbolically, a disjunction is represented as "A ∨ B." Here's the truth table for a disjunction: - If either A or B is true, then the statement "A or B" is true. - If both A and B are false, then the statement "A or B" is false. In the context of the exercise, the statement "Paris is not in France or London is not in England" exemplifies a logical disjunction. It necessitates that at least one part of the negation is true. Recognizing how disjunctions function is crucial for interpreting the results of logical negations and understanding broader logical systems.