Problem 5 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Which of the following is a statement? (A) May you live long! (B) May God bless you! (C) The sun is a star. (D) Hurrah! We have won the match.

Step-by-Step Solution

Answer

Option (C) "The sun is a star." is a statement.

Step 1: Understand what a statement is

A statement is a declarative sentence that is either true or false but not both. It conveys information that can be verified.

Step 2: Analyze Option (A)

The sentence "May you live long!" is a wish, not a declarative statement. It cannot be classified as true or false.

Step 3: Analyze Option (B)

The sentence "May God bless you!" is also a wish, similar to option (A). It is not a declarative statement and cannot be true or false.

Step 4: Analyze Option (C)

The sentence "The sun is a star." is a declarative sentence. It conveys factual information that can be verified and is indeed true, making it a statement.

Step 5: Analyze Option (D)

"Hurrah! We have won the match." includes an interjection and while the latter part may be declarative, the expression of joy is not a complete statement in itself.

Key Concepts

Declarative Sentences

In mathematical logic, declarative sentences are vital in distinguishing statements from other types of sentences. A declarative sentence is a type of sentence that conveys information and makes a statement about something. Its main characteristic is that it can be either true or false, but not both simultaneously. This is why these are typically utilized in logic, mathematics, and computer science, as they provide clear assertions that can be assessed.

• Declarative sentences are straightforward and provide facts or assertions.
• Unlike questions or wishes, they don't express emotions or requests.
• They must have a clear subject and predicate structure.

Without the precise nature of declarative sentences, determining the truth value would be impossible, making them indispensable in logical reasoning.

Truth Value

The truth value of a sentence refers to whether that statement is true or false. This binary evaluation is fundamental in both mathematical logic and real-world problem-solving, providing a basis for determining the validity of arguments.

• A statement can carry one of two truth values: true (T) or false (F).
• The statement "The sun is a star" is true, as it conveys accurate factual information.
• Wishes or emotions, such as "Hurrah!" or "May you live long," cannot have a truth value since they do not convey factual information.

Understanding truth values enables us to apply logical operations and verify information through a structured process, often utilizing truth tables and logical proofs.

Verification of Statements

Verification is the process of determining whether a given declarative sentence carries a truth value of true or false. This is a crucial aspect of mathematical logic, allowing us to confirm or refute assertions based on logical reasoning and evidence.

• Verification often involves checking facts and using evidence, such as scientific knowledge or reliable data sources.
• The statement "The sun is a star" can be verified through astronomical observation and scientific consensus.
• Non-declarative sentences, like exclamations or wishes, cannot be verified because they do not communicate factual content.

By thoroughly verifying statements, we can ensure that the information we use in decision-making and reasoning is accurate and reliable. This ultimately supports logical consistency and sound conclusions.