g10-02-03 · Core topics
All or at least one?
Can a statement be existentially true but universally false?
1 MIN · PREDICT ON PAPER
For x²≥k or x²>k, compare a finite integer domain with all real numbers.
Discuss or write first. No typed answer is required to continue.
All: false; exists: true. Domain: finite integers.
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
All requires every allowed input; existence requires one. Enlarging the domain can change these conclusions differently. Finite displayed cases suggest a conjecture; the justification below states why it holds under the stated conditions.
Conditions: The finite domain is {−n,…,n}, n≥1. The alternative domain is R.
Mathematical justification: The minimum square is 0. On the finite domain the maximum is n²; on R squares are unbounded. These bounds decide all and existence exactly.
1 MIN · EXIT CHECK
Over R, compare ∀x:x²≥1 and ∃x:x²≥1.
Check after answering
The universal claim is false at 0; the existential claim is true at 1.
Teacher guidance and extension
Before: Squares and quantifier language
Ask: Ask pupils to distinguish a witness from a complete proof over R.
Watch for: Checking the plotted real inputs is not checking every real number.
Challenge: Find k for which the finite domain has no witness but R has one.
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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