What is the name of this rule? $ c\lor (a \land \lnot c)= c \lor a $

Question

What is the name of this rule? $$ c\lor (a \land \lnot c)= c \lor a $$

I know if $c$ is true then the right side does not matter. If $c$ is false, then it all depends on $a$. But what is the rule called?

Answer 1

Together with its dual form $c \land (a \lor \neg c)$, I have long referred to this equivalence as Reduction: the way I talk about it, the $c$ term ‘reduces’ the $a \land \neg c$ term to just $a$.

While Reduction can be derived from other principles, when doing boolean algebra I find it to be super useful to be able to do this in 1 step: it is syntactically easy to remember and apply, it is semantically fairly intuitive (certainly the dual version is), and it comes up many, many times when doing boolean algebra. So, I think it definitively deserves its own name and should be a fixture in any list of boolean algebra equivalences.

In fact, during the many years I have used it I thought for sure I saw this equivalence named as Reduction somewhere, but doing a quick search now I find that nothing comes up other than myself using it in answer to other MSE questions. Hey, if I turn out to be the first to name it I’ll gladly take the credit, LOL

EDIT: I wonder if I adopted the usage of ‘reduction’ as it is used in the Davis-Putnam algorithm. There, you ‘reduce’ a clause (or clause set) with regard to a unit literal by taking out any complement of that literal from the clause (or clause set). E.g. the clause $\{ a, \neg c \}$ (which of course can be seen to represent $a \lor \neg c $) would reduce to just $\{ a \}$ with regard to $c$.