Types of Statements

Different types of statements explained with clear definitions and simple examples used in mathematical reasoning.

1. Idea of Different Types of Statements

In mathematical reasoning, statements come in several forms depending on how they express information. Even though every statement must have a definite truth value, the way it is written may vary — some talk about specific objects, some talk about all objects, and some describe conditions. Knowing the type of a statement helps in understanding how to reason with it.

1.1. Why classification helps

Different statements behave differently when negated, combined, or used in proofs. Recognizing the structure of a statement makes it easier to apply logical rules correctly.

2. Declarative (Assertive) Statements

A declarative statement simply states a fact or claim, and it is either true or false. Most basic mathematical sentences fall into this category.

2.1. Examples

3. Open Statements

An open statement contains a variable whose value is not fixed. Because the truth of the statement depends on the value of the variable, it is not a statement until the variable is specified.

3.1. Examples

  • "x > 5"
  • "n is a perfect square"

These become statements only when the variables are assigned particular values.

4. Universal Statements

A universal statement makes a claim about all objects of a certain kind. These statements often begin with words like "for all" or "every".

4.1. Examples

5. Existential Statements

An existential statement asserts that at least one object with a certain property exists. These statements usually include phrases like "there exists" or "for some".

5.1. Examples

  • "There exists a number whose square is 5."
  • "There exists an integer that is even and prime."

6. Conditional Statements (Implications)

A conditional statement expresses a logical connection between two statements. It has the form "If P, then Q", where P is called the hypothesis and Q is the conclusion.

This type of statement is central to proofs and logical deduction.

6.1. Examples

  • "If a number is even, then it is divisible by 2."
  • "If x > 3, then x² > 9."

7. Bi-conditional Statements

A bi-conditional statement asserts that two statements are true together — each implies the other. It has the form "P if and only if Q", often written as "P ⇔ Q".

7.1. Examples

  • "A number is even if and only if it is divisible by 2."
  • "Two triangles are congruent if and only if they have the same three sides."

8. Notes and Observations

Some key points about types of statements:

  • Declarative statements have fixed truth values.
  • Open statements need variable values to become actual statements.
  • Universal and existential statements involve quantifiers.
  • Conditional and bi-conditional statements express relationships between ideas.