Turn failed universal claims into precise witnesses without overclaiming.
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Try
A submission table has one row per learner and one column per task. A filled cell means that learner submitted that task. A claim about every learner having some submission is a claim that every row contains at least one filled cell.
Predict
Predict what a failed row must look like. Missing one task is not enough: the learner could have submitted another. Negation means the entire claim is false, not that every word is replaced by its opposite. Keep both domains fixed while negating.
Experiment
Sketch three learners and two tasks. Fill both cells in the first row, no cells in the second, and one cell in the third. Identify a witness against “every row has a filled cell.” Then add one filled cell to the empty row and determine whether the original statement now holds. The learner action is to locate a witness, not to calculate a quality score.
Understand
Worked argument: negate “For every real x there exists a real y with y²=x.” Its negation says there exists a real x such that for every real y, y²≠x. The choice x=−1 verifies the negation because every real square is nonnegative. We need one failing target x, and we must rule out all proposed y for that target.
The invalid diagnosis “no real number is a square” is far stronger and false: 4 is the square of 2. Similarly, one learner with no submission does not imply that everyone submitted nothing.
Apply
Translate each practice claim into ordinary language as well as quantifiers. A strong response identifies a witness for an existential part and explains how every value in a universal part is handled. Distinguish an empty row from an empty column: they answer different questions.
Continue
Depth: if there are no enrolled learners, “every enrolled learner submitted some task” is true by the meaning of universal quantification; its negation is false because no failing learner exists. If learners exist but there are no tasks, the original claim is false. Explain these cases without changing the domains.
Apply · explain · revise
Practice and reasoning
Drafts save locally as you type. Reveal worked solutions when useful; the site remembers that exposure. Automatic matches check the requested answer, while written arguments require review against the rubric.
Negate: Every enrolled learner submitted at least one task.
Learners Ada, Bo and Cy submitted respectively tasks {1,2}, {}, and {2}. How many learners witness failure of the claim?
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
Write and justify the negation of: for every learner l there exists a task t with Submitted(l,t).
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Explain why 'No learner submitted any task' is too strong as a negation, using the Ada/Bo/Cy example.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Reflect and continue
Before marking this lesson complete, explain one key idea without the worked solution and identify a condition that matters. Marking complete records participation, not independent mastery.
