g10-02-04 · Core topics
Negating a quantified statement
Does negating “all are greater” mean “all are smaller”?
1 MIN · PREDICT ON PAPER
Predict the negation of ∀x:x²>k and ∃x:x²>k on [−n,n].
Discuss or write first. No typed answer is required to continue.
Negation: ∃x: x² ≤ k; True
2 MIN · EXPLAIN AND REVISE
Which condition does your claim need?
Compare at least two cases. Write a reason, then check the explanation together.
Reveal explanation and conditions
Negation switches all and exists and replaces > by ≤. Equality belongs to the negation. Finite displayed cases suggest a conjecture; the justification below states why it holds under the stated conditions.
Conditions: The domain is the entire real interval [−n,n], with n>0.
Mathematical justification: On this interval 0≤x²≤n², with both bounds attained. ∀x:x²>k holds iff k<0; ∃x:x²>k holds iff k<n². Their negations are the complementary conditions.
1 MIN · EXIT CHECK
Negate “every real x has x²>0”.
Check after answering
There exists a real x with x²≤0; x=0 is a witness.
Teacher guidance and extension
Before: Quantifiers and strict inequalities
Ask: Use k=0 to expose the missing equality in an incorrect negation.
Watch for: Not all is not the same as none.
Challenge: Negate an inequality with ≥ and justify the new strict endpoint.
Suggested timing: predict 1 min, compare 3 min, explain 2 min, exit check 1 min. Observations can suggest a claim; they do not prove it for every input.
Optional local notebook
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