Negation of the statement: '√5 is an integer or 5 is irrational' is: (A) √5 is irrational or 5 is an

Q: Why does the 'or' change to 'and' when we negate the statement?

Think of it this way: for an 'or' statement to be true, you only need one part to be correct. Therefore, to prove the whole statement false, you must ensure that *neither* part is correct. If you only negate one, the 'or' condition might still be satisfied by the other part. Thus, you need both to be false, which is exactly what the 'and' operator enforces.

Q: Is the negation of 'is irrational' simply 'is rational'?

Technically, yes, in the context of real numbers. However, in formal logic, the negation of a predicate $P$ is simply 'not $P$'. Writing '5 is not irrational' is logically equivalent to '5 is rational', but for the purpose of these problems, sticking to the 'not' form is the safest way to ensure you don't accidentally change the meaning of the statement.

Q: Does the order of the statements matter in the final answer?

In logic, the conjunction (AND) is commutative, meaning $A \land B$ is the same as $B \land A$. However, it is best practice to maintain the original order of the components to avoid confusion and ensure your answer matches the provided options perfectly.

Q: What if I just negate the individual parts but forget to change the 'or'?

That is the most common trap in JEE logic questions! If you keep the 'or', you are essentially saying that at least one of the conditions must be false. But the negation of 'at least one is true' is 'none are true'. By keeping the 'or', you fail to capture the requirement that *both* must be false.

Q: Are there any other laws like De Morgan's that I should memorize?

Absolutely! You should be familiar with the Law of Double Negation ($\neg(\neg p) \equiv p$), the Law of Implication ($\neg(p \implies q) \equiv p \land \neg q$), and the distributive laws. Mastering these is like having a Swiss Army knife for logical reasoning.

Question

Negation of the statement: '$\sqrt{5}$ is an integer or 5 is irrational' is:

Options:

(A) $\sqrt{5}$ is irrational or 5 is an integer.

(B) $\sqrt{5}$ is not an integer or 5 is not irrational.

(C) $\sqrt{5}$ is an integer and 5 is irrational.

(D) $\sqrt{5}$ is not an integer and 5 is not irrational.

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Welcome, future engineer! Today, we are stepping away from the heavy machinery of calculus and the abstract fields of electromagnetism to explore the very foundation of all mathematical thought: Logic.

You might think, 'It is just a simple negation, why do I need a masterclass for this?' But here is the secret: the JEE Advanced exam doesn't just test your ability to calculate; it tests your ability to think with absolute precision.

When we negate a statement, we are not just flipping words; we are performing a surgical operation on the truth value of a proposition.

### Deconstructing the Compound Statement

Imagine you are presented with a complex claim: '$\sqrt{5}$ is an integer or 5 is irrational'. To analyze this, we must first break it down into its atomic components.

Let us define $p$ as the statement '$\sqrt{5}$ is an integer' and $q$ as the statement '5 is irrational'. The entire sentence is then represented by the logical disjunction $p \lor q$.

In the world of logic, the 'or' operator ($\lor$) is inclusive. It tells us that the statement is true if $p$ is true, if $q$ is true, or if both are true.

To negate this, we are looking for the condition under which the entire statement $p \lor q$ is false. This is where the beauty of De Morgan's Law comes into play.

### The Power of De Morgan's Law

De Morgan's Law is one of the most elegant symmetries in mathematics. It states that the negation of a disjunction is the conjunction of the negations:

$$
eg(p \lor q) \equiv 
eg p \land 
eg q$$

Think about the physical intuition here. If I tell you, 'I will go to the park OR I will go to the library,' and I turn out to be a liar, what must be true?

It means I did NOT go to the park AND I did NOT go to the library. The 'or' collapses into an 'and' because the only way to invalidate an 'or' is to deny every single possibility it offers.

### Executing the Negation

Now, let us apply this to our specific problem. We have our components:

1. $p$: $\sqrt{5}$ is an integer. Its negation, $
eg p$, is '$\sqrt{5}$ is not an integer'.
2. $q$: 5 is irrational. Its negation, $
eg q$, is '5 is not irrational'.

By applying the transformation $
eg p \land 
eg q$, we combine these two negated statements with the 'AND' operator. The result is: **'$\sqrt{5}$ is not an integer AND 5 is not irrational'**.

### Why This Matters

I know it feels simple, but this is the same logic used in designing complex digital circuits and writing robust computer algorithms. If you are writing code to check if a user's input is invalid, you are constantly using these logical transformations.

If you fail to flip the operator from 'or' to 'and', your code will have a 'bug'—a logical flaw that could lead to catastrophic system failures in real-world engineering.

So, as you look at the final result, appreciate the symmetry. We started with a statement that was 'loose' (an 'or' statement), and through the rigorous application of De Morgan's Law, we arrived at a statement that is 'strict' (an 'and' statement).

You have successfully navigated the logical landscape. Keep this precision in your toolkit, and you will find that even the most daunting JEE problems become clear, logical paths to the truth.

Animated Solution for Mathematics - Sets and Relations: Negation of the statement: ' $5$ is an integer or 5 is irrational' is:

Select Answer:

Visualized Solution

Identify Components $p$ and $q$

Identify the Logical Connective

Define the Goal

Apply De Morgan's Law

Negate Statement $p$

Negate Statement $q$

Transform the Operator

Combine for Final Answer

The Sigma Insight: Types of Sets and Set Operations

Analyzing the Setup

Deconstructing the Compound Statement

The Power of De Morgan's Law

Executing the Negation

Why This Matters

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Animated Solution for Mathematics - Sets and Relations: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Negation of the statement: '5​ is an integer or 5 is irrational' is: