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## DeMorgan: ¬(A ∧ B) ⫤⊨ ¬A ∨ ¬B

`\$\leftmodels\models\$' means `is logically equivalent to', so for now `has the same truth table as'.
A \$\leftmodels\models\$ ¬¬A
¬(A ∧ B) \$\leftmodels\models\$ (¬A ∨ ¬B)
¬(A ∨ B) \$\leftmodels\models\$ (¬A ∧ ¬B)
A → B \$\leftmodels\models\$ ¬A ∨ B
¬(A → B) \$\leftmodels\models\$ ¬(¬A ∨ B) \$\leftmodels\models\$ A ∧ ¬B
Here's a useful equivalence: double negations cancel out (at least in logic).

A ⫤⊨ ¬¬A

is logically equivalent to

i.e. has the same truth table as

 A ¬A ¬¬A ​T F T ​F T F
 ¬(A ∧ B) ⫤⊨ (¬A ∨ ¬B) ¬(A ∨ B) ⫤⊨ (¬A ∧ ¬B) A → B ⫤⊨ ¬A ∨ B ¬(A → B) ⫤⊨ ¬(¬A ∨ B) ⫤⊨ A ∧ ¬B
 A B A ∧ B ¬(A ∧ B) ¬A ¬B ¬A ∨ ¬B ​T T T F F F F ​T F F T F T T ​F T F T T F T ​F F F T T T T
 A B ¬(A ∨ B) ¬A ∧ ¬B ​T T F F ​T F F F ​F T F F ​F F T T
 A B A → B ¬A ∨ B ​T T T T ​T F F F ​F T T T ​F F T T
 A B A → B ¬(A → B) ¬(¬A ∨ B) A ∧ ¬B ​T T T F F F ​T F F T T T ​F T T F F F ​F F T F F F
3.19
4.15--18
7.1--7.2, *7.3--7.6
3.19
4.31