Problem 8 - A Complete Resource Book in Mathematics for JEE Main

Question

Negation of \(" 2+3=5\) and \(8<10^{\prime \prime}\) is (A) \(2+3 \neq 5\) and \(8<10\) (B) \(2+3=5\) and \(8 \times 10\) (C) \(2+3 \neq 5\) or \(8<10\) (D) None of these

Step-by-Step Solution

Answer

D) None of these.

Step 1: Understand Negation

Negation involves changing the truth value of a statement to its opposite. The negation of a statement "P and Q" is "not P or not Q".

Step 2: Negating the First Statement

The statement "2+3=5" is true. The negation of this statement is "2+3 ≠ 5."

Step 3: Negating the Second Statement

The statement "8<10" is true. The negation of this statement is "8 ≥ 10."

Step 4: Constructing the Negated Statement

The original statement is "2+3=5 and 8<10." The negation requires us to write "2+3 ≠ 5 or 8 ≥ 10." However, in the context of the given options, we look for any statement that shows a negation which could be interpreted as "2+3 ≠ 5" or some alteration regarding the comparison "<".

Step 5: Evaluating Options

Option A and Option C include parts of these negations, but neither fits completely logically under the transformation. Option D, "None of these," implies neither matches entirely correctly the logical negation of the given statement under normal rules.

Key Concepts

Negation

Negation plays a crucial role in mathematical logic by inverting the truth value of logical statements. It's all about changing a positive, true statement to a negative, false one, or vice versa. To negate a statement like "P and Q," you must understand that negation transforms it to "not P or not Q." This rule captures the essence of logical complementation in compound statements.

When you negate individual statements, it's important to understand the base truth. For instance, the statement "2 + 3 = 5" is true. Therefore, its negation reads as "2 + 3 ≠ 5," shifting the truth to false. Similarly, negating the true statement "8 < 10" yields "8 ≥ 10," flipping the truth value from true to false. The process of negation ensures that every positive or true assertion has its counter false statement or invalid proposition.

Truth Value

In the realm of logic, each statement is evaluated based on its truth value, which determines whether it's true or false. The transformation of a statement's truth value under negation uncovers interesting insights into logical structures.

• A statement with a truth value of "true" becomes false when negated.
• A statement with a truth value of "false" turns true upon negation.

The statement "2 + 3 = 5" has an initial truth value of true, hence its negated form "2 + 3 ≠ 5" ends up having a truth value of false. Conversely, since the statement "8 < 10" starts as true, its negation "8 ≥ 10" acquires a truth value of false.

Understanding truth values is fundamental in deducing the outcome of compound statements. Through logical operations, such as negation, we effectively challenge stated facts or assumptions to explore their opposites.

Logical Statement

A logical statement is a declarative sentence that holds a truth value of either true or false. Logical statements are the foundation of logical reasoning and provide a structured framework for assessing truths and derivations in mathematics.

Common examples include mathematical equations like "2 + 3 = 5," which is definitively true. Another key aspect of logical statements is their composition; they can often combine using logical connectors such as "and," "or," and "not." These connectors allow for the construction of complex propositions, enhancing logical evaluative processes.

• "And" combines two statements; both must be true for the overall statement to be true.
• "Or" allows for either statement to be true for the overall statement to be true.
• "Not" negates the statement's original truth value.

Understanding how logical statements work, especially in compound configurations, aids in navigating complex logical scenarios, leading to clearer and more accurate conclusions.