Logical Statement Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Logical Statement.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A declarative sentence that has exactly one definite truth value — either true (T) or false (F), never both and never neither.

A logical statement is any claim that can be judged definitively as true or false — questions, commands, and paradoxes are not statements.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Every statement has exactly one truth value: T or F. This binary nature is what makes logical reasoning systematic and checkable.

Common stuck point: 'This statement is false' is a paradox—it's not a proper statement.

Sense of Study hint: Try assigning T or F to the sentence. If you can do exactly one, it is a valid statement. If neither works or both work, it is not.

Worked Examples

Example 1

easy
Classify each as a statement (proposition) or not: (a) 3 + 5 = 8 (b) 'Close the door.' (c) x > 2.

Solution

  1. 1
    (a) 3 + 5 = 8 is a declarative sentence that is true. It is a statement.
  2. 2
    (b) 'Close the door' is a command, not declarative. It is not a statement.
  3. 3
    (c) x > 2 has an unspecified variable, so its truth value is undetermined. It is an open sentence, not a statement.

Answer

\text{(a) Statement (true), (b) Not a statement, (c) Not a statement (open sentence)}
A logical statement must be a declarative sentence with a definite truth value—either true or false, not both and not indeterminate.

Example 2

medium
Negate the statement: 'All prime numbers are odd.'

Example 3

medium
Classify each as a statement or not: (a) '7 + 3 = 11' (b) 'Is 5 prime?' (c) 'Every square is a rectangle.'

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Determine the truth value of each: (a) 7 is a prime number. (b) 0 is a natural number. (c) Every square is a rectangle.

Example 2

easy
Which of the following are logical statements: (a) '7 is prime', (b) 'Open the window', (c) 'What is 2+5?', (d) '3^2 = 10'?
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