logic: derive a formula using laws

Question

Let's say I have the following formula: $$(A\wedge\neg C)\vee(B\wedge C)\vee(A\wedge B).\tag{1}$$ It is easy to show following: $$(A\wedge\neg C)\vee(B\wedge C)\vee(A\wedge B)\Leftrightarrow (A\wedge\neg C)\vee(B\wedge C).$$ What about deriving $(A\wedge\neg C)\vee(B\wedge C)$ from $(1)$ using Boolean algebra laws? (laws)

Answer 1

Using a more algebraic notation, one gets \begin{align} A\bar C + BC + AB &= A\bar C + BC + AB(C + \bar C) && \text{(Law $C + \bar C = 1$)}\\ &=A\bar C + BC + ABC + AB\bar C && \text{(Distributivity)} \\ &=(A + AB)\bar C + (B + AB)C&& \text{(Distributivity)} \\ &=A\bar C + BC&& \text{(Absorption)} \end{align}