Simple Logical Simplification - Distributive Law I can't seem to understand the hopefully simple step of getting from :
$(a \land b \land c) \lor (a \land b \land \lnot c)$
to:
$(a \land b) \land (c \lor \lnot c)$
This step is in the answers and only has '(distribution)' written next to it, however even when breaking this down into several parts with Distributive Laws, I got nothing close to the expected solution (most likely my fault rather than teacher's). So any help explaining this step would very much be appreciated. Thank you!
Apply the distributive law to $d \wedge (c\vee \neg c)$ where $d=a\wedge b$. Then you get $$\begin{array}{rcl}d\wedge (c\vee \neg c)&amp;=&amp;(d\wedge c )\vee (d\wedge \neg c)\\&amp;=&amp;((a\wedge b)\wedge c)\vee ((a \wedge b)\wedge \neg c)\\&amp;=&amp;(a\wedge b\wedge c)\vee (a\wedge b \wedge \neg c). \end{array}$$Hope this helps!
It might help to put it into words:
I will use $TRUE/FALSE$ notation.
$A,B,C$ are $TRUE$ or $A,B$, NOT $C$ are $TRUE$. In either case $A$ and $B$ are $TRUE$.
Then either $C$ is $TRUE$ or NOT $C$ is $TRUE$ since this is an or statement and may be either one.