Question

Which of the following sentences is/are statement(s)?
(i) 10 is less than 5.
(ii) All rational numbers are real numbers.
(iii) Today is a sunny day.

• A. (i), (ii) and (iii)
• B. (i) and (ii) only
• C. (i) and (iii) only
• D. (ii) and (iii) only
• E. (i) only

Hint:

Logic Tip: Don't confuse "false" with "not a statement". A statement can be entirely wrong (like $1+1=5$), as long as it isn't an opinion or an open-ended question!

Answer 1

The Correct Option is B

Concept:
In mathematical logic, a "statement" (or proposition) is a declarative sentence that is definitively either true or false, but not both. Sentences that rely on subjective opinions, changing contexts (like time or location), or variables without specified values do not qualify as statements.

Step 1: Evaluate sentence (i).
Sentence: "10 is less than 5." This is a declarative sentence that makes a specific factual claim. The claim is mathematically false. Because it has a definitive truth value (False), it is a valid statement.

Step 2: Evaluate sentence (ii).
Sentence: "All rational numbers are real numbers." This sentence declares a mathematical property. By definition, the set of rational numbers is a subset of the real numbers. Because this is definitively True, it is a valid statement.

Step 3: Evaluate sentence (iii).
Sentence: "Today is a sunny day." The truth of this sentence depends entirely on the location of the speaker and the current date. It is subjective and context-dependent, lacking a universal truth value. Therefore, it is not a statement.

Step 4: Compile the valid statements.
Based on our evaluations in the previous steps: Sentence (i) is a statement. Sentence (ii) is a statement. Sentence (iii) is not a statement.

Step 5: Select the corresponding option.
Since only sentences (i) and (ii) meet the criteria of mathematical statements, we look for the option that includes exactly these two. Hence the correct answer is (B) (i) and (ii) only.