Grade 12 · Reading & language

Distinguish necessary and sufficient conditions

Use the exact direction of a stated rule to distinguish a guarantee from a requirement, without reversing an implication.

8 authored questions + worked example

Worked example

The rule says: If a badge has a blue triangle, it opens this gate. No rule about other badges is supplied.

What does the blue triangle establish under this rule?

Answer & why

Answer: It is sufficient for opening; the text does not establish that it is necessary.

The rule guarantees opening for a blue-triangle badge but does not exclude other opening badges.

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All questions

All 8 authored questions for this skill.

Question 1

In this invented system, every complete form is approved. A specific form was not approved. Treat the rule as exceptionless.

Which inference is valid?

  1. Every incomplete form must be rejected.
  2. Approval never occurs for any form.
  3. That form was complete.
  4. That form was not complete.
A hint

If A then B guarantees B when A holds; B only if A makes A required. Do not silently turn either rule into an equivalence.

Answer & why

Answer: That form was not complete.

If it were complete, the rule would require approval. Nonapproval therefore rules out completeness.

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Question 2

A model says a signal is accepted exactly when tests A and B both succeed. A succeeds but B fails.

What follows within this model?

  1. The signal is accepted because one test succeeds.
  2. Every signal fails both tests.
  3. The signal is not accepted.
  4. The two tests are irrelevant.
A hint

If A then B guarantees B when A holds; B only if A makes A required. Do not silently turn either rule into an equivalence.

Answer & why

Answer: The signal is not accepted.

Exactly when defines the joint requirement; one success does not satisfy both tests.

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Question 3

The rule says: If a badge has a blue triangle, it opens this gate. No rule about other badges is supplied.

What does the blue triangle establish under this rule?

  1. It is sufficient for opening; the text does not establish that it is necessary.
  2. Every badge that opens must have it.
  3. It proves that no badge can open.
  4. It is necessary, but cannot be sufficient.
A hint

If A then B guarantees B when A holds; B only if A makes A required. Do not silently turn either rule into an equivalence.

Answer & why

Answer: It is sufficient for opening; the text does not establish that it is necessary.

The rule guarantees opening for a blue-triangle badge but does not exclude other opening badges.

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Question 4

The notice says: A visitor is admitted only if the visitor has a ticket. It gives no promise that every ticket holder will be admitted.

What role does a ticket have in the stated rule?

  1. It is necessary, but admission is not guaranteed by it alone.
  2. It is neither relevant nor required.
  3. It is sufficient, but not necessary.
  4. It guarantees admission and no other conditions are possible.
A hint

If A then B guarantees B when A holds; B only if A makes A required. Do not silently turn either rule into an equivalence.

Answer & why

Answer: It is necessary, but admission is not guaranteed by it alone.

Only if makes the ticket a requirement; the notice does not make it a complete guarantee.

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Question 5

This scoring protocol says a result is labelled pass if and only if its score is at least 80.

How does a score of at least 80 relate to the pass label?

  1. It is sufficient but the protocol allows lower passing scores.
  2. It has no stated connection to passing.
  3. It is necessary but explicitly insufficient.
  4. It is both necessary and sufficient in this protocol.
A hint

If A then B guarantees B when A holds; B only if A makes A required. Do not silently turn either rule into an equivalence.

Answer & why

Answer: It is both necessary and sufficient in this protocol.

If and only if supplies both directions: the score qualifies, and passing requires that score.

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Question 6

An idealized rule states: Every sample containing marker X reacts. A particular sample does not contain marker X. Nothing is said about other causes of reaction.

Which conclusion is justified?

  1. Every reacting sample must contain X.
  2. The sample certainly cannot react.
  3. The rule alone does not determine whether this sample reacts.
  4. The sample certainly must react.
A hint

If A then B guarantees B when A holds; B only if A makes A required. Do not silently turn either rule into an equivalence.

Answer & why

Answer: The rule alone does not determine whether this sample reacts.

The rule gives a sufficient condition, not an exclusive cause; lacking it does not settle the result.

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Question 7

Assume this rule always holds: If the indicator is lit, the circuit is enabled. A check establishes that the circuit is not enabled.

What follows from the rule and this check?

  1. The circuit has no physical components.
  2. Every enabled circuit has a lit indicator.
  3. The indicator is not lit.
  4. The indicator must be lit.
A hint

If A then B guarantees B when A holds; B only if A makes A required. Do not silently turn either rule into an equivalence.

Answer & why

Answer: The indicator is not lit.

A lit indicator would imply enabled; the established not-enabled result rules out a lit indicator.

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Question 8

The protocol says a trial may proceed only if both reviewers approve. Both reviewers have approved, but no other permission rule is stated.

What can be concluded about proceeding?

  1. One reviewer must now withdraw approval.
  2. The stated requirement is met, but that sentence alone does not guarantee permission.
  3. The requirement has not been met.
  4. Permission is guaranteed because only if means if and only if.
A hint

If A then B guarantees B when A holds; B only if A makes A required. Do not silently turn either rule into an equivalence.

Answer & why

Answer: The stated requirement is met, but that sentence alone does not guarantee permission.

Both approvals satisfy a necessary condition. The text does not say it is sufficient.

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