1. ¬(a & b) ∨ b = ¬a ∨ ¬b ∨ b = ¬a ∨ (¬b ∨ b) = ¬a ∨ 1 = 1
2. ¬( (a ∨ ¬b) → (c & ¬a) ) = ¬( ¬(a ∨ ¬b) ∨ (c & ¬a) ) =
(a ∨ ¬b) & ¬(c & ¬a) = (a ∨ ¬b) & (¬c ∨ a) =
(a & ¬c) ∨ (a & a) ∨ (¬b & ¬c) ∨ (a & ¬b) =
(a & ¬c) ∨ a ∨ (¬b & ¬c) ∨ (a & ¬b) =
a & (¬c ∨ 1 ∨ ¬b) ∨ (¬b & ¬c) =
a & 1 ∨ (¬b & ¬c) = a ∨ ¬b & ¬c
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Answers & Comments
1. ¬(a & b) ∨ b = ¬a ∨ ¬b ∨ b = ¬a ∨ (¬b ∨ b) = ¬a ∨ 1 = 1
2. ¬( (a ∨ ¬b) → (c & ¬a) ) = ¬( ¬(a ∨ ¬b) ∨ (c & ¬a) ) =
(a ∨ ¬b) & ¬(c & ¬a) = (a ∨ ¬b) & (¬c ∨ a) =
(a & ¬c) ∨ (a & a) ∨ (¬b & ¬c) ∨ (a & ¬b) =
(a & ¬c) ∨ a ∨ (¬b & ¬c) ∨ (a & ¬b) =
a & (¬c ∨ 1 ∨ ¬b) ∨ (¬b & ¬c) =
a & 1 ∨ (¬b & ¬c) = a ∨ ¬b & ¬c