Problem 47 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

Negation of the statement: \(\sqrt{5}\) is an integer of 5 is irrational is: \(\quad\) (a) \(\sqrt{5}\) is not an integer or 5 is not irrational (b) \(\sqrt{5}\) is not an integer and 5 is not irrational (c) \(\sqrt{5}\) is irrational or 5 is an integer. (d) \(\sqrt{5}\) is an integer and 5 is irrational

Step-by-Step Solution

Answer

(b) 'sqrt(5) is not an integer and 5 is not irrational' is the negation.

Step 1: Understand the Given Statement

The original statement is: 'sqrt(5) is an integer OR 5 is irrational'. This is a compound statement with the logical operator 'or'.

Step 2: Determine the Negation

According to De Morgan's laws, to negate a compound statement with 'or', you negate each part and change 'or' to 'and'. So, the negation of 'P or Q' is 'not P and not Q'.

Step 3: Apply De Morgan's Laws

Negate each part of the original statement. The negation of 'sqrt(5) is an integer' is 'sqrt(5) is not an integer'. The negation of '5 is irrational' is '5 is not irrational'. So, the negation becomes 'sqrt(5) is not an integer AND 5 is not irrational'.

Step 4: Identify the Correct Option

Compare the negated statement with the given choices. Option (b) 'sqrt(5) is not an integer and 5 is not irrational' matches the negated statement.

Key Concepts

Compound Statements

In logic, compound statements are formed when we combine two or more statements using logical connectors such as 'and', 'or', 'not', etc. These compound statements allow us to express complex ideas that are made up of simpler propositions. For example, consider the statement \'\( \sqrt{5} \text{ is an integer OR 5 is irrational} \)\'. This is a compound statement because it combines two distinct ideas:

• '\( \sqrt{5} \text{ is an integer} \)' and
• '5 is irrational'

...using the logical connector 'OR'.
Understanding how these statements combine helps in analyzing their truth values and is essential for applications in mathematics and computer science.
The way we connect propositions within a compound statement affects how we negate them, as shown later in De Morgan’s laws. Be sure to grasp how each part interacts within the whole.

Logical Negation

Logical negation involves reversing the truth value of a statement.
If a statement is true, its negation is false, and vice versa. Negating compound statements often involves applying specific rules, such as De Morgan's Laws.
These laws state that:

• Negation of 'P OR Q' is 'NOT P AND NOT Q'.
• Negation of 'P AND Q' is 'NOT P OR NOT Q'.

For the exercise, negating the compound statement \'\( \sqrt{5} \text{ is an integer OR 5 is irrational} \)\' involves:

• Turning 'OR' into 'AND'.
• Negating both parts:
• \'\( \sqrt{5} \text{ is not an integer} \)', and
• '5 is not irrational'.

Using these transformations, we ensure the correct representation of the original statement's negation.
This process highlights how logic can systematically handle oppositional values.

Irrational Numbers

Irrational numbers are real numbers that cannot be expressed as a simple fraction or ratio of two integers.
Unlike rational numbers that can be neatly written as \( \frac{a}{b} \) where \(a\) and \(b\) are integers, irrational numbers go on forever without repeating. Some common examples include \(\sqrt{2}\), \(\pi\), and \(e\).
This continuous, non-repeating nature makes them fascinating subjects of study within mathematics.

• The number \( \sqrt{5} \) itself falls into the category of irrational numbers.
• It cannot be precisely written out as a fraction, aligning with its placed role in examples for negations.

Understanding irrational numbers is not just an exercise in classification but extends to applications in solving equations where precise decimal values are unnecessary or impossible.
Next time you see a square root that doesn’t simplify to a clean whole number, you are likely dealing with another of these captivating quantities.