g10-02-01 · Core topics
A counterexample to an implication
When does p⇒q fail?
1 MIN · PREDICT ON PAPER
Test p: x>c and q: x>0; predict the one truth-value combination that refutes p⇒q.
Discuss or write first. No typed answer is required to continue.
This input is not a counterexample to p⇒q.
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
A counterexample needs p true and q false. A case with p false does not refute the implication. Finite displayed cases suggest a conjecture; the justification below states why it holds under the stated conditions.
Conditions: x and c are real numbers. A displayed test at one x is distinct from the claim for every x.
Mathematical justification: By the definition of implication, only true⇒false fails. For example c<0 permits a point c<x≤0 as a refutation.
1 MIN · EXIT CHECK
Does x=−1 refute x>−2 ⇒ x>0?
Check after answering
Yes: −1>−2 but −1 is not positive.
Teacher guidance and extension
Before: Inequalities and truth values
Ask: Ask whether each example refutes, supports locally, or leaves a universal claim unproved.
Watch for: Finding q true does not establish p; the converse is a separate statement.
Challenge: Choose c for which the forward implication always holds but the converse fails.
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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