Help with Boolean Expression Simplification

Question

I know that

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

and

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

are supposed to be equal but I'm unsure how to simplify the first boolean expression past cancelling the first and last terms due to the identity rule.

Answer 1

Consider that:

$$\begin{align} (a\land b\land c)\lor (a\land \lnot b\land c) & = ((a\land c)\land b)\lor((a\land c)\land\lnot b) & \textsf{association and commutation} \\ & = (a\land c)\land (b\lor\lnot b) & \textsf{distribution} \\ & = (a\land c) \land \top & \textsf{complementation} \\ & = a\land c & \textsf{conjunctive identity} \end{align}$$

Now can you use this lesson to further simplify your statement?

Answer 2

$(a∧¬b∧c)∨(a∧b∧c)$ = $(a∧c)∧(¬b∨b)$ = $(a∧c)$

use this to get to your answer