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Solve both parts and justify the conditions used. Part A: Give an integer counterexample. Part B: Count the possible integer witnesses.

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 / Transfer#Your turn

Solve both parts and justify the conditions used. Part A: Give an integer counterexample. Part B: Count the possible integer witnesses.

\begin{gathered}\text{A: }\forall n\in\mathbb Z,\ n^2>n\\\text{B: }-7<x<7\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: Keep the domain explicit and distinguish all values from one witness.

Hint 2

B: Keep the domain explicit and distinguish all values from one witness.

Worked solution

Part A reasoning
Keep the domain explicit and distinguish all values from one witness.
Calculate or simplify this relation.
$0^2=0\not>0$
One witness of failure is sufficient.
Part B reasoning
Keep the domain explicit and distinguish all values from one witness.
Calculate or simplify this relation.
$(6)-(-6)+1=13$
Existence needs one witness; counting asks for all admissible witnesses.

A: n=0 is a counterexample. B: The requested value is 13.

Checks and common pitfalls: One witness of failure is sufficient. Existence needs one witness; counting asks for all admissible witnesses.

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.