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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
TFT
FTF
¬(A ∧ B)    ⫤⊨     (¬A ∨ ¬B)
¬(A ∨ B)⫤⊨(¬A ∧ ¬B)
A → B⫤⊨¬A ∨ B
¬(A → B)⫤⊨¬(¬A ∨ B)    ⫤⊨    A ∧ ¬B
ABA ∧ B¬(A ∧ B)¬A¬B¬A ∨ ¬B
TTTFFFF
TFFTFTT
FTFTTFT
FFFTTTT
AB¬(A ∨ B)¬A ∧ ¬B
TTFF
TFFF
FTFF
FFTT
ABA → B¬A ∨ B
TTTT
TFFF
FTTT
FFTT
ABA → B¬(A → B)¬(¬A ∨ B)A ∧ ¬B
TTTFFF
TFFTTT
FTTFFF
FFTFFF
3.19
4.15--18
7.1--7.2, *7.3--7.6
3.19
4.31