Q.For the following statement, determine whether an inclusive "Or" or exclusive "Or" is used. Give reasons for your answer: Students can take French or Sanskrit as their third language.

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