Negation — Definition, Meaning & Examples
Negation is the logical opposite of a statement. If a statement is true, its negation is false, and if a statement is false, its negation is true.
In logic, the negation of a statement is a new statement, written (read "not "), that has the opposite truth value of . Negation is a unary logical operation, meaning it applies to a single statement rather than combining two statements. The negation of is true when is false, and false when is true.
Key Formula
Where:
- = the negation symbol, read as "not"
- = the original statement
Worked Example
Problem: Write the negation of each statement and determine its truth value: (a) "All squares are rectangles." (b) "7 is an even number."
Step 1: Identify statement (a) and its truth value. "All squares are rectangles" is a true statement.
p$: All squares are rectangles. \quad p \text{ is } \textbf{true}
Step 2: Write the negation of statement (a). To negate "all," you say "not all" (which is the same as "some ... are not"). Since the original is true, the negation is false.
\sim p$: Not all squares are rectangles. \quad \sim p \text{ is } \textbf{false}
Step 3: Identify statement (b) and its truth value. "7 is an even number" is a false statement.
q$: 7 is an even number. \quad q \text{ is } \textbf{false}
Step 4: Write the negation of statement (b). Since the original is false, the negation is true.
\sim q$: 7 is not an even number. \quad \sim q \text{ is } \textbf{true}
Answer: The negation of "All squares are rectangles" is "Not all squares are rectangles" (false). The negation of "7 is an even number" is "7 is not an even number" (true).
Why It Matters
Negation is fundamental to writing proofs in geometry and discrete mathematics. When you prove something by contradiction, you start by assuming the negation of what you want to prove and show that it leads to an impossible conclusion. Negation is also essential for forming contrapositives, which are logically equivalent to the original conditional statement and often easier to prove.
Common Mistakes
Mistake: Negating "all" by saying "none."
Correction: The negation of "all A are B" is "some A are not B" — not "no A are B." For example, the negation of "all cats are black" is "some cats are not black," not "no cats are black."
Mistake: Thinking the negation of a conditional is .
Correction: The negation of is actually (the original hypothesis is true AND the conclusion is false). Writing gives you the inverse, which is a different statement.