# 3.3  Algebraic Properties

We conclude this chapter with a set of identities pertaining to special cases and compositions of logical operations.

The first two lemmas are immediate consequences of the definitions.

Lemma 3.3.1   (logand-x-0,logior-x-0,logxor-x-0) For all ,

 (a) ; (b) ; (c) .

Lemma 3.3.2   (logand-self,logior-self,logxor-self) For all and ,

 (a) ; (b) ; (c) .

PROOF: By Definition 3.2.1,

All of the remaining results of this section may be derived in a straightforward manner from Lemmas 3.2.5, 3.1.14, and 2.3.22.

Lemma 3.3.4   (logand-x-m1,logior-x-m1,logxor-x-m1) For all ,

 (a) ; (b) ; (c) .

PROOF: Suppose . By Lemma 3.1.14, for all

and it is readily seen by exhaustive computation that this implies . It follows from Lemma 2.3.22 that . A similar argument applies to (b).

The proofs the remaining lemmas are sufficiently similar to that of Lemma 3.3.5 that they may be safely omitted.

Lemma 3.3.6   (logand-commutative,logior-commutative,logxor-commutative) For all , , and ,

 (a) ; (b) ; (c) .

Lemma 3.3.7   (logand-associative,logior-associative,logxor-associative) For all , , and ,

 (a) ; (b) ; (c) .

Lemma 3.3.8   (logior-logand,logand-logior,log3) For all , , and ,

 (a) ; (b) ; (c) .

Lemma 3.3.9   (logxor-rewrite,lognot-logxor) For all and ,

 (a) ; (b) .

David Russinoff 2017-08-01