Problem 3 - A Complete Resource Book in Mathematics for JEE Main

Question

Which of the following is not a statement? (A) Roses are red. (B) New Delhi is in India. (C) Every square is a rectangle. (D) Alas! I have failed.

Step-by-Step Solution

Answer

Option D is not a statement.

Step 1: Understand the Definition of a Statement

A statement is a declarative sentence that is either true or false, but not both. It makes an assertion.

Step 2: Analyze Each Option

Let's look at each option to see if it fits the definition of a statement. - Option A: 'Roses are red.' is a declarative sentence stating something about roses. It can be true or false. - Option B: 'New Delhi is in India.' is also declarative and can be verified as true or false. - Option C: 'Every square is a rectangle.' is a declarative sentence that is always true, fitting the statement definition. - Option D: 'Alas! I have failed.' is an exclamatory sentence and does not assert anything verifiable as true or false.

Step 3: Determine Which Option is Not a Statement

Among the given options, Option D does not fit the definition of a statement because it is an exclamation, and it does not have a truth value that can be determined.

Key Concepts

Statements in Mathematics

In mathematics, a statement is a fundamental building block for logical reasoning. It is a specific kind of sentence, known as a declarative sentence, that must convey clear and unambiguous information. This type of sentence makes a direct assertion that something is either true or false. Such statements are essential for proofs, modeling, and calculations.
By establishing propositions as statements, mathematicians can then determine their truth values and develop logical sequences that lead to conclusions.
Here are some characteristics of mathematical statements:

• They should be declarative.
• They must be true or false but cannot be both.
• They are used to build reasoning in mathematical arguments.

Without statements, it would be challenging to construct valid logical arguments or to engage effectively with mathematical proofs.

Declarative Sentences

The term 'declarative sentence' refers to a sentence that makes a clear, emphatic assertion. It conveys information directly and is designed to be understood without ambiguity. This is crucial in mathematics, where precision is key.
Declarative sentences lay the groundwork for mathematical statements by explicitly stating facts that can be evaluated for truthfulness. Let's explore some features of declarative sentences:

• They state an idea or fact directly.
• They are crucial for conveying mathematical assertions.
• They should leave no room for ambiguity.

For example, in the exercise, "Roses are red" and "New Delhi is in India" are declarative. These sentences offer clear assertions that can be independently checked for truth.

Truth Values

Truth values are fundamental in assessing the validity of statements. A truth value indicates whether a statement is "true" or "false". This binary evaluation is key to mathematical logic, as it allows for the construction of logical frameworks and proofs.
In the context of the exercise, each option is assessed for its truth value. For instance, the statement "New Delhi is in India" is true, and "Every square is a rectangle" is also verifiably true.
Significant aspects of truth values include:

• They determine the validity status of a statement.
• Truth values are binary – "true" or "false".
• They are essential for logical operations and comparisons.

The use of truth values facilitates clarity and rigor in mathematical arguments, ensuring that conclusions are derived from valid premises.