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Solve both parts and justify the conditions used. Part A: Repair this proposed implication. Part B: Disprove: a positive product forces both factors to be positive.

Read the idea, work independently, then explain what changed.

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TOPIC 01

Sets and commonly used logic: mixed assessment

A mixed assessment: identify the method, justify it and revise your reasoning.

What you will be able to explain

Connect the chapter skills without relying on the order of the exercises.

Try it. Leave your reasoning visible.

Use one hint at a time. A correction explains what changed, not just the final answer.

01 / Foundation#Your turn

Solve both parts and justify the conditions used. Part A: Repair this proposed implication. Part B: Disprove: a positive product forces both factors to be positive.

\begin{gathered}\text{A: }xy=xz\Rightarrow y=z\\\text{B: }ab>0\end{gathered}

Use the domain, units and sampling assumptions stated in the question.
Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

A: Test both directions and seek a counterexample when a direction fails.

Hint 2

B: Test both directions and seek a counterexample when a direction fails.

Worked solution

Part A reasoning
Test both directions and seek a counterexample when a direction fails.
Calculate or simplify this relation.
$x(y-z)=0\Rightarrow x=0\text{ or }y=z$
Cancellation requires a nonzero factor.
Part B reasoning
Test both directions and seek a counterexample when a direction fails.
Calculate or simplify this relation.
$(-8)(-9)=72>0$
A single admissible counterexample disproves the universal claim.

A: It holds if x≠0; without this condition it can fail. B: Choose a=-8, b=-9.

Checks and common pitfalls: Cancellation requires a nonzero factor. A single admissible counterexample disproves the universal claim.

Reasoning checklist · self / teacher assessment

Solve part A with its stated restrictions.
Solve part B using an appropriate representation.
Give the reasoning and check conditions in both parts.

Think first. Reveal a hint when the class is ready.

Teacher preparation and assessment

Question sequence

Ask students to name the relevant condition before calculating.

Board plan

Compare valid methods and annotate their conditions.

Anticipated thinking

A correct final value may still hide a missing assumption.

Assessment checklist

Check the method, conditions, reasoning and interpretation separately.