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

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

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

Step 1: Understanding Statements

A statement is a declarative sentence that is either true or false, but not both. It expresses a complete idea that can be evaluated for truth.

Step 2: Evaluating Each Option

Let's evaluate each option to determine if it is a statement: - (A) 'May you live long!' is a wish; wishes are not statements because they cannot be true or false. - (B) 'May God bless you!' is a blessing; like wishes, blessings are not statements because they cannot be evaluated as true or false. - (C) 'The sun is a star.' is a declarative sentence that can be evaluated as true, making it a statement. - (D) 'Hurrah! We have won the match.' is an exclamation, which again cannot be true or false.

Step 3: Choosing the Correct Option

Out of all the options, only (C) 'The sun is a star.' is a statement because it is the only one that is a declarative sentence that can be evaluated for truth.

Key Concepts

Statements

In mathematical logic, a statement is a fundamental concept. It refers to a sentence that declares something about a subject. Statements must be either true or false; they cannot be both at the same time. This is a critical attribute that distinguishes statements from other types of sentences like questions, commands, or exclamations. For example:

• The sentence "The earth orbits the sun" is a statement because it can be determined to be true.
• Conversely, "Are you coming to the party?" is not a statement since it poses a question and cannot clearly be categorized as true or false.

Statements are used to express definitive information that can be logically analyzed. When you encounter a sentence in math or logic problems, the first step is to assess if it fulfills the criteria of being a statement.

Declarative Sentence

Declarative sentences are crucial in forming logical statements. These are sentences that make declarations or assertions about a fact or condition. It should present a complete idea and be structured in a way that allows for immediate verification in terms of truth or falsehood. Here are some characteristics:

• Declarative sentences often start with a subject and a verb. For example, "Birds can fly." is a declarative sentence.
• They are typically direct and straightforward in their presentation.
• If a sentence includes a wish, command, or an emotional expression, it is not considered declarative, even if it includes factual information.

Using declarative sentences accurately allows us to construct clear, evaluable statements that support reasoning and truth assessment in logical discourse.

Truth Evaluation

Truth evaluation is the process of determining whether a statement is true or false. This concept is foundational in math and logic. A statement's truth value is crucial because it dictates how it can be used in reasoning and further argumentation. Here are key aspects:

• The statement "The moon is made of cheese" can be evaluated and found to be false based on empirical evidence.
• Truth values are commonly represented in binary form: true (T) or false (F). This simplicity allows for integration into logical systems and the construction of truth tables.
• The context of a statement can affect its truth value. For instance, "It is raining" is only true if it is indeed raining at the time and place in question.

Engaging with truth evaluation helps clarify thinking and underpins the logical consistency of arguments and problem-solving techniques.