Logic, First-Order Representation and Entailment Short Notes for GATE DA: Concepts, Formulas, Worked Examples & Practice

Logic, First-Order Representation and Entailment short notes for GATE DA: 2 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.

Quick Recap: Translation and Validity

Final Checklist

1. Translation Keys

All $A$ are $B$ : $\forall x (A (x) ⟹ B (x))$
Some $A$ are $B$ : $\exists x (A (x) \land B (x))$
No $A$ are $B$ : $\forall x (A (x) ⟹ \neg B (x))$

2. Negation

$\neg\forall \equiv \exists\neg$
$\neg\exists \equiv \forall\neg$

3. Validity Check

Try to falsify. If you cannot, it is valid.
Watch out for empty domains (standard FOL assumes non-empty).

4. Common Pitfall

$\exists x (A (x) ⟹ B (x))$ is rarely the intended meaning of "Some A are B".

Summary: Entailment and Reasoning Checklist

Final Checklist for Entailment

1. Definitions

$X ⊨ Y$ : In all models where $X$ is true, $Y$ is true.
$X ⟹ Y$ : A formula that evaluates to true or false within a specific model.

2. Proving & Disproving

Prove: Use logical equivalences or deduction.
Disprove: Construct a single counter-model where $X$ is True and $Y$ is False.

3. Structural Rules

$X ⊨ Y ⟹ X \land Z ⊨ Y$ (Monotonicity)
$X ⊨ Y ⟹ X ⊨ Y \lor Z$ (Weakening)

4. FOL Manipulations

Contrapositive: $\forall x (P (x) ⟹ Q (x)) \equiv \forall x (\neg Q (x) ⟹ \neg P (x))$
Negation flip: $\neg\forall x P (x) \equiv \exists x \neg P (x)$

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Logic, First-Order Representation and Entailment Short Notes for GATE DA: Concepts, Formulas, Worked Examples & Practice

Logic, First-Order Representation and Entailment short notes for GATE DA: 2 study cards covering concepts, formulas, shortcuts and exam traps, plus solved pra

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