Problem 6 - A Complete Resource Book in Mathematics for JEE Main

Question

Which of the following is not a statement? (A) Please do me a favour. (B) 2 is an even integer. (C) \(2+1=3\). (D) The number 17 is prime.

Step-by-Step Solution

Answer

Option (A) is not a statement.

Step 1: Understanding the concept of a statement

A statement is a sentence that is either true or false, but not both. It must have a truth value that can be determined clearly.

Step 2: Analyzing Option (A)

"Please do me a favour." is a sentence that is a request or command rather than a statement with a truth value. It is not true or false since it is asking someone to perform an action.

Step 3: Analyzing Option (B)

"2 is an even integer." is a statement because it can be verified as true. The integer 2 is indeed even.

Step 4: Analyzing Option (C)

"2 + 1 = 3." is a mathematical statement with a truth value. This particular statement can be verified as true because the sum of 2 and 1 is indeed 3.

Step 5: Analyzing Option (D)

"The number 17 is prime." is a statement because it can be evaluated as true. A prime number is one that has no divisors other than 1 and itself, and 17 fits this definition.

Step 6: Determining the Non-statement

Based on the analysis, Option (A) is not a statement because it does not have a truth value, whereas options (B), (C), and (D) are statements with identifiable truth values.

Key Concepts

Statements in Logic

In mathematical logic, the concept of a **statement** is fundamental. A statement is a declarative sentence that can be classified as either true or false, but not both. These kinds of sentences hold a truth value which helps in logical reasoning. For example:

• "2 is an even integer." - This can be verified as it is either true or false.
• "The glass is on the table." - Given correct conditions, this also holds truth value.

However, not every sentence qualifies as a statement. Sentences that are questions, commands, or requests do not have truth values. For instance, "Please do me a favour." is a request, so it lacks a clear truth value to determine its status as a statement.

Truth Values

The notion of **truth values** in logic refers to the classification of statements as either true or false. Every statement in logic must have one or the other. This clear distinction is crucial for logical evaluation and reasoning.
Here are a few key points:

• True: When a statement holds or logically aligns with facts, like "2 + 1 = 3," it is true.
• False: If the statement contradicts known facts, it is deemed false.

Truth values allow us to systematically approach and solve problems logically. Determining the truth value of each statement allows mathematicians and logicians to build accurate arguments and proofs.

Prime Numbers

Prime numbers are central to number theory and play a critical role in mathematics. A **prime number** is defined as a natural number greater than 1 that has no divisors other than 1 and itself. Prime numbers are the building blocks of the entire number system because any integer greater than one is either a prime or can be factored into primes.
Here are some examples and properties:

• Examples: 2, 3, 5, 7, 11, 13, and 17 are prime numbers.
• Properties: Numbers like 4, 6, 8 are not prime as they have divisors other than 1 and themselves.

Identifying whether a number is prime is crucial in fields such as cryptography, since the security of many encryption algorithms depends on large prime numbers.

Even Integers

An even integer is a number that is divisible by 2 without any remainder. Even integers form one of the simplest categories of numbers in mathematics. Here’s how you can identify them:

• Definition: An integer is even if it can be expressed in the form 2n, where n is an integer.
• Examples: 0, 2, 4, 6, 8, and 10 are typical examples of even numbers.
• Properties: Even numbers always end with 0, 2, 4, 6, or 8.

In arithmetic operations, the sum or difference of two even numbers is always even, and an even number multiplied by any integer always results in an even number. Recognizing even integers and understanding their properties is foundational for various branches of mathematics.