A question related to lattice theory.

Question

I am asked to show that in a Boolean Algebra $$(a' \lor b') \lor (a \land b \land c') \;=\; (b \land c') \lor (a' \lor b')$$ My question is - Is it absolutely okay to show this using truth table formally? Or else I have to use the definition ? But I guess the steps would get long and time consuming. Any help from experts is welcome.

Answer 1

Let's examine the left hand side.

$\begin{align}(a'\vee b') \vee (a\wedge b\wedge c') & = (a\wedge b)'\wedge ((a\wedge b)\wedge c') & \textsf{de Morgan's Negation}\\ & = & \textsf{Absorption}\\ & = & \textsf{de Morgan's Negation} \\ & = & \textsf{Association and Commutation}\\ & = & \textsf{Absorption (reverse)} \\ & = (b\wedge c') \vee (a'\vee b') & \textsf{Association and Commutation} \end{align}$