Mathematically Acceptable Statements

Typically, a statement is a sentence that is not an order or an exclamatory sentence or a question. Generally, a statement can be in any of the following conditions, such as always true, always false or ambiguous. There are two situations that make the statement ambiguous. The first situation is that “we cannot decide if the statement is always true or always false”. The second situation that leads to the ambiguity of a statement is that “the statement is subjective”. It means that the statement is true for some people and not true for other people. In this article, we will learn the definition of mathematically acceptable statements with many solved examples.

What are Mathematically Acceptable Statements?

In Mathematics, a statement is only acceptable or valid, if it is either true or false. Such a statement is called a mathematically acceptable statement. In other words, a statement is true, if it is always true, otherwise, a statement is a false statement. A statement can be a simple statement or a compound statement.

For example, 3+4=7 is always true. Hence, it is a true statement.

Similarly, 3+4=8 is always false. Hence, it is a false statement.

Note: A mathematical statement cannot be ambiguous.

Now, let us understand the different types of statements with the help of examples.

Example: The sun rises in the east.

The given statement is always true. Hence, it is a true statement. This is because the sun rises in the east no matter where we live.

Example: There are 8 days in a week.

The given statement is a false statement since there are 7 days in a week.

Example: It is raining here.

The given statement is an ambiguous statement because it is not clear where “here” is.

Mathematically Acceptable Statement Examples

Check whether the given statements are mathematically acceptable statements:

1. The sum of interior angles of a triangle is 180°.
2. The product of two odd integers is always even.
3. Rahul is a kind boy.

Solutions:

1. The statement “the sum of interior angles of a triangle is 180°” is a true statement. The statement is always true and it is proved by the angle sum property of a triangle. Hence, it is a mathematically acceptable statement.
2. The statement “the product of two odd integers is even” is a false statement. Because, the product of two odd integers is always odd. For example, the product of 3 and 3 is 9, which is an odd number. Hence, the given statement is a mathematical statement.
3. The statement “Rahul is a kind boy” is an ambiguous statement, since the statement is subjective. As the given statement is ambiguous, it is not a mathematically acceptable statement.

Practice Problems

Identify the kind of statement which are given below:

1. The product of two even natural numbers is an even number.
2. It is very hot today.
3. Every odd number greater than 1 is a prime number.
4. For every real number, 2x is greater than x.
5. If a quadrilateral has all its sides equal, then it is a rectangle.

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