In logical reasoning, a statement is a sentence that is either true or false, but not both.
Negation
The negation of a statement reverses its truth value, usually by inserting the word "not". If a statement $p$ is true, its negation $\sim p$ is false, and vice versa. For example, the negation of "$7$ is a prime number" (true) is "$7$ is not a prime number" (false).
Key formula
Compound statements: "$p$ and $q$" is true only when both $p$ and $q$ are true. "$p$ or $q$" is true when at least one of $p$, $q$ is true.
Compound statements
Two statements can be joined by "and" or "or" to form a compound statement. Use the rules above to decide its truth value.
Worked example
Let $p$: "$3>2$" (true) and $q$: "$5$ is even" (false). Then "$p$ and $q$" is false (both must be true). "$p$ or $q$" is true (at least one is true). The negation of $q$, "$5$ is not even", is true.
Remember
- "And" needs both parts true; "or" needs at least one.
- Negation always flips true to false and false to true.
- A statement must be definitely true or false; questions and commands are not statements.