<s>[INST] How can I simplify my boolean expression further I have the following boolean expression that I want to simplify
$$B\cdot D+ \overline{A\cdot B\cdot D} +  \overline{B}\cdot C\cdot  \overline{D}$$
Here is what I have been able to due so far
$$B\cdot D+ \overline{A} + \overline{B} + \overline{D} + \overline{B} \cdot  \overline{D}\cdot C$$
I know that the answer is suppose to be
$$\overline{A} + \overline{B}\cdot B $$
How can I simplify my initial expression any further thank you very much any help.
 [/INST] Factorising
$$\overline{A}+B\cdot D + \overline{B}+\overline{D} \cdot (1+\overline{B}\cdot C)$$
$$=\overline{A}+B\cdot D + \overline{B}+\overline{D} \cdot 1$$
$$=\overline{A}+B\cdot D + \overline{B}+\overline{D}$$
Then by the absorption law,
$$B\cdot D + \overline{B}=D + \overline{B}$$
So this simplifies to
$$=\overline{A}+D + \overline{B}+\overline{D}$$
$$=\overline{A} + \overline{B}+D+\overline{D}$$
$$=\overline{A} + \overline{B}+1$$
$$=1$$
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