Simplification - Boolean Algebra Expression

Question

Can somebody please help me simplify this expression.

$(b \lor c) \lor (b \land a) \lor (a \land c)$

Thank you.

Answer 1

Might be easiest with a truth table. It would simplify to $(b \lor c)$

Answer 2

First note note that because $\lor$ is an Associative operator (meaning that $p \lor (q \lor r) = (p \lor q) \lor r$, you can effectively remove the parentheses.

So, your expression can simply be written as:

$b \lor c \lor (b \land a) \lor (a \land c)

Now, here is a handy equivalence:

Absorption

$p \lor (p \land q) = p$ (the $p$ term 'absorbs the $p \land q$ term)

In your statement, we thus see that the $b$ term absorbs the $b \land a$ term, and that the $c$ term absorbs the $a \land c$ term, leaving you with $b \lor c$

Formally:

$$(b \lor c) \lor (b \land a) \lor (a \land c)$$

$$\overset{Association}{=}$$

$$b \lor c \lor (b \land a) \lor (a \land c)$$

$$\overset{Absorption}{=}$$

$$b \lor c$$

Finally, if you don't have a one-step Absorption in the set of rules you are allowed to use, then this will probably work instead:

$$p \lor (p \land q)$$

$$\overset{Identity}{=}$$

$$(p \land \top) \lor (p \land q)$$

$$\overset{Distribution}{=}$$

$$p \land (\top \lor q)$$

$$\overset{Annihilation}{=}$$

$$p \land \top $$

$$\overset{Identity}{=}$$

$$p$$