<s>[INST] Boolean algebra transformation I read online that
$${\displaystyle (A\cdot {\overline {B}})+({\overline {A}}\cdot B)\equiv (A+B)\cdot ({\overline {A}}+{\overline {B}})}$$
and I can verify this, but I'm not quite sure how to basically take the left-hand side and transform it into the right-hand side. Any help would be appreciated, thank you.
 [/INST] *

*As right hand can be transformed in way:
$$(A+B)\cdot ({\overline {A}}+{\overline {B}}) \equiv  \\
 A \cdot \overline {A}+A \cdot\overline {B} + B  \cdot  \overline {A} +  B  \cdot  \overline {B } \equiv \\
 A\cdot {\overline {B}}+{\overline {A}}\cdot B$$
and knowing, that equivalence works in both direction, you can now write this equivalences in opposite direction to reach desired.


*For second way let's use $\overline{A\cdot B}=\overline{A} + \overline {B}$ and $\overline{A+ B}=\overline{A} \cdot \overline {B}$:
$$A\cdot {\overline {B}}+{\overline {A}}\cdot B \equiv \overline { \overline{A}+B}+\overline {A+\overline {B}} \equiv \\
\overline {(\overline{A}+B) \cdot (A+\overline {B})} \equiv \overline {A\cdot B + \overline{A} \cdot \overline {B}} \equiv \overline{A\cdot B} \cdot \overline{\overline{A} \cdot \overline {B}} \equiv \\
(A+B)\cdot ({\overline {A}}+{\overline {B}})
$$
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