Translating "Not" into Symbols
A free Logic lesson from the “Statements and Negation” unit, with a worked example and practice problems including step-by-step solutions.
The exact negation holds in every case where the original fails and in no case where the original holds. Over real numbers, not greater than means less than or equal to. Not all means at least one not; not some means none. Check the boundary and avoid strengthening the claim.
What you'll learn
- Translate not statements using ¬p notation
- Explain the idea in plain English before using symbols
- Use examples, non-examples, or counterexamples to check the reasoning
Worked example
Problem. Negate x > 40 over the real numbers.
- The negation includes the boundary and all lower values.
Answer: x <= 40
Lesson preview: this page shows 5 practice items. The interactive activity contains 6 practice items and 6 quiz items (multiple choice). Counts include repeated prompts and variants.
Practice problems
1. Negate x > 25 over the real numbers.
Choices: x < 25 · x >= 25 · x = 25 · x <= 25
Show solution
- The negation includes the boundary and all lower values.
Answer: x <= 25
2. Negate x <= 25.
Choices: x < 25 · x != 25 · x > 25 · x >= 25
Show solution
- A value fails to be at most the bound exactly when it exceeds it.
Answer: x > 25
3. Negate x = 25.
Choices: x < 25 · x != 25 · x = -25 · x > 25
Show solution
- Either a smaller or a larger value makes the equality false.
Answer: x != 25
4. Negate “Every drawer is locked.”
Choices: At least one drawer is not locked · No drawer is locked · At least one drawer is locked · Every drawer is locked
Show solution
- One unlocked drawer is enough to make the universal claim false.
Answer: At least one drawer is not locked
5. Negate “Some files are missing.”
Choices: Some files are not missing · Every file is missing · At least one file is missing · No files are missing
Show solution
- Some asserts existence.
- Its negation rules out every instance.
Answer: No files are missing
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