I have the following propositional formula:

$$\lnot A \land (\lnot A \lor \lnot B) \land (\lnot A \lor C) \land (\lnot B \lor C)$$

I can see and "explain" that the middle two terms in brackets are redundant, so the formula simplifies to:

$$\lnot A \land (\lnot B \lor C)$$

I can't, however, figure out how to get there formally using the associativity laws of $\land$ and $\lor$, de Morgan's laws, etc.

Can anyone please give some help with those formal steps? Thanks in advance.


Note that $P\land(P\lor Q)\equiv P$, so $$\lnot A \land (\lnot A \lor \lnot B) \land (\lnot A \lor C) \land (\lnot B \lor C)\equiv\lnot A \land(\lnot A\lor(\lnot B\land C))\land (\lnot B \lor C)$$$$\equiv\lnot A\land(\lnot B \lor C).$$

  • $\begingroup$ That's exactly the sort of property I was looking for, thanks very much. $\endgroup$ – Michael Sep 23 '13 at 7:35
  • $\begingroup$ @Michael This property sometimes gets called "absorption." $\endgroup$ – Doug Spoonwood Sep 23 '13 at 14:29

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