I am starting discrete mathematics and it came to me that distributive lattices has 2 equicalent representations for their definition:
- $a\wedge (b\vee c) = (a \wedge b)\vee (a\wedge c)$
- $a\vee (b\wedge c) = (a \vee b)\wedge (a\vee c)$
However, I couldn't find proof for this equivalence. I am using "Introduction to discrete structures for computer science and engineering" by Preparata, FRANCO. If you can help me with an insight or at least some place I could research about the matter, I'd be very grateful.
Using the first formulation, you get \begin{align} (a\vee b) \wedge (a\vee c) &= ((a\vee b)\wedge a) \vee ((a\vee b) \wedge c)\\ &= a \vee ((a \vee b) \wedge c) \tag{absorption}\\ &= a \vee (c \wedge (b \vee a))\\ &= a \vee ((c \wedge b) \vee (c \wedge a)) \tag{first formulation}\\ &= a \vee (a \wedge c) \vee (b \wedge c)\\ &= a \vee (b\wedge c). \tag{absorption} \end{align}
The converse is analogous.