Derivation Rules

Covers equivalence rules, inference rules, and how they are applied in proof sequences.

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Views #: 106

Questions #: 17

Time: 20 minutes

Pass Score: 80.0%

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Categories of Derivation Rules

1 pts

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What are the two categories of derivation rules for propositional logic?

Hypotheses and conclusions

Inference rules and contradictions

Equivalence rules and truth tables

Equivalence rules and inference rules

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Equivalence Rules

1 pts

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What do equivalence rules allow?

Hypotheses to be added automatically.

Individual wffs to be rewritten.

New wffs to be derived from previous wffs.

Inference rules to be reversed.

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Inference Rules

1 pts

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What do inference rules allow?

Individual wffs to be rewritten.

The addition rule to be reversed.

Any wff to be substituted anywhere.

New wffs to be derived from previous wffs.

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Direction of Rules

1 pts

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How do equivalence and inference rules differ regarding direction?

Equivalence rules work in both directions; inference rules do not.

Inference rules work in both directions; equivalence rules do not.

Neither works in both directions.

Both work in both directions.

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Commutative Rule

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Which rule states \(P∨Q\) is equivalent to \(Q∨P\) and \(P∧Q\) is equivalent to \(Q∧P\)?

Commutative

Implication

De Morgan

Associative

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Associative Rule

1 pts

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Which rule states \((P∨Q)∨R\) is equivalent to \(P∨(Q∨R)\)?

Commutative

Definition of equivalence

Associative

De Morgan

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De Morgan's Laws

1 pts

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Which rule states \((P∨Q)'\) is equivalent to \(P'∧Q'\) and \((P∧Q)'\) is equivalent to \(P'∨Q'\)?

Commutative

Implication

De Morgan's Laws

Double negation

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Implication Rule

1 pts

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Which rule states \(P→Q\) is equivalent to \(P'∨Q\)?

Associative

Definition of equivalence

Implication

De Morgan

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Double Negation

1 pts

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Which rule states \(P\) is equivalent to \((P')'\)?

Simplification

Implication

Double negation

Commutative

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Definition of Equivalence

1 pts

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Which rule states \(P↔Q\) is equivalent to \((P→Q)∧(Q→P)\)?

Addition

Implication

Definition of equivalence

De Morgan

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Modus Tollens

1 pts

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From \(P→Q\) and \(Q'\), what can be derived by modus tollens?

\(Q'\)

\(Q\)

\(P\)

\(P'\)

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Conjunction Rule

1 pts

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From \(P\) and \(Q\), what can be derived by conjunction?

\(P→Q\)

\(P'\)

\(P∧Q\)

\(P∨Q\)

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Simplification Rule

2 pts

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From \(P∧Q\), what can be derived by simplification? Select all that apply.

\(Q\)

\(P→Q\)

\(P∨Q\)

\(P\)

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Addition Rule

1 pts

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From \(P\), what can be derived by addition?

\(P'\)

\(P∨Q\)

\(Q\)

\(P∧Q\)

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Reversing Addition

1 pts

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Can the addition rule be reversed?

Yes, from \(P∨Q\) we can infer \(Q\).

Yes, from \(P∨Q\) we can infer \(P∧Q\).

Yes, from \(P∨Q\) we can infer \(P\).

No, from \(P∨Q\) we cannot infer either \(P\) or \(Q\).

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Modus Ponens Pattern Match

1 pts

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If \((A→B)∨C\) and \(A\) are hypotheses, why does modus ponens not apply?

\(A\) is not a hypothesis.

\(C\) is false.

The main connective of \((A→B)∨C\) is disjunction, not implication.

The main connective of \((A→B)∨C\) is implication.

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Justification in a Proof

1 pts

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In a proof sequence, what is the role of the justification for a step?

It is a required part of the proof sequence.

It replaces the derivation rule.

It makes the conclusion true.

It is not a required part of the proof sequence, but it confirms that the step is legitimate.

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Year 11 Year 12

Categories of Derivation Rules

Correct Answer

Explanation

Equivalence Rules

Correct Answer

Explanation

Inference Rules

Correct Answer

Explanation

Direction of Rules

Correct Answer

Explanation

Commutative Rule

Correct Answer

Explanation

Associative Rule

Correct Answer

Explanation

De Morgan's Laws

Correct Answer

Explanation

Implication Rule

Correct Answer

Explanation

Double Negation

Correct Answer

Explanation

Definition of Equivalence

Correct Answer

Explanation

Modus Tollens

Correct Answer

Explanation

Conjunction Rule

Correct Answer

Explanation

Simplification Rule

Correct Answer

Explanation

Addition Rule

Correct Answer

Explanation

Reversing Addition

Correct Answer

Explanation

Modus Ponens Pattern Match

Correct Answer

Explanation

Justification in a Proof

Correct Answer

Explanation

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