Q.For the following statement, determine whether an inclusive "Or" or exclusive "Or" is used. Give reasons for your answer: The school is closed if it is a holiday or a Sunday.

What is a Mathematical Statement?

Think about everyday statements. "It is raining." "The train is late." "Mumbai is the capital of India." Each of these is either true or false — that's the core idea. A mathematical statement is exactly the same, but it deals with mathematical objects.

A mathematical statement is a declarative sentence that is either true or false, but not both.

That's it. The entire definition rests on two pillars: the sentence must declare something (not ask, not command, not exclaim), and it must have a definite truth value.

The Two Conditions, Unpacked

Condition 1: It must be a declarative sentence.

"Solve $x^2 = 4$" is a command — not a statement. "What is the square root of 4?" is a question — not a statement. "Oh, what a beautiful proof!" is an exclamation — not a statement. Only a sentence that asserts something qualifies.

Condition 2: It must be either true or false, and not both.

This is where most confusion happens. Consider:

"The number 2 is even."

True. So it's a statement.

"The number 2 is odd."

False. Still a statement — false statements are still statements.

"This sentence is false."

If it's true, it's false. If it's false, it's true. This is the liar paradox — it's neither consistently true nor consistently false. So it is not a mathematical statement.

The Tricky Part: Open Sentences

Here's where students often slip. Look at this:

"$x + 3 = 7$"

Is this a statement? You cannot say — because $x$ is not fixed. If $x = 4$, the sentence is true. If $x = 5$, it's false. The truth value depends on $x$. This is called an open sentence (or a predicate). It becomes a statement only when you either:

• Substitute a specific value: "$4 + 3 = 7$" (true statement)
• Quantify it: "There exists an $x$ such that $x + 3 = 7$" (true statement)

Examples to Cement the Idea

Why This Matters

Every theorem, every proof, every logical argument in mathematics is built from statements. When you prove something, you are showing that one statement (the conclusion) follows necessarily from other statements (the hypotheses). If you cannot clearly identify what is and isn't a statement, you cannot construct or follow a proof.

Quick Check

Which of these are mathematical statements?

1. $5$ is a factor of $12$.
2. $x$ is a factor of $12$.
3. For every integer $x$, $x$ is a factor of $12$.
4. Solve $x^2 - 4 = 0$.
5. $2$ is the only even prime number.

Answers: 1 (false statement), 3 (false statement), 5 (true statement). 2 is an open sentence. 4 is a command.