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@@ -99,3 +99,33 @@ Truth Table: - with a truth table - by (3) 3. Use table of Common Equivalences (see link above) + +--- + +# Using the Table of Common Equivalences + +- Ex. Show `(p^q)->(p->q)` is a tautology without a truth table + - We are goint to solve this using a proof + - **Goal**: `(p^q)->(p->q)≡T` + +|Proof: (p^q)->(p->q)|Reason| +|--------------------|------| +|≡(p^q)->(┓p∨q) |Implication| +|≡┓(p^q)∨(┓p∨q) |Implication| +|≡(┓p∨┓q)∨(┓p∨q) |De Morgan| +|≡┓p∨(┓q∨(┓p∨q)) |Associative| +|≡┓p∨(┓q∨(q∨┓p)) |Commutative| +|≡┓p∨((┓q∨q)∨┓p)) |Associative| +|≡┓p∨(T∨┓p)) |Negation| +|≡┓p∨T |Domination| +|≡T |Domination| + +- Why use this method? + - This method is useful for compound propositions with many propositions + +- `n` propositions -> truth table has 2^n rows + - e.g 5 propositions -> truth table has 32 rows (!) + +--- + +[1.4 ->](1.4.md) |