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Unformatted text preview: d to multiplication and addition modulo 2, respectively. One curious property of this algebra is that every element is its own additive inverse: ˆ Á ´ µ ˆ ¼.
Aside: Who, besides mathematicians, care about Boolean rings? Every time you enjoy the clarity of music recorded on a CD or the quality of video recorded on a DVD, you are taking advantage of Boolean rings. These technologies rely on error-correcting codes to reliably retrieve the bits from a disk even when dirt and scratches are present. The mathematical basis for these error-correcting codes is a linear algebra based on Boolean rings. End Aside. We can extend the four Boolean operations to also operate on bit vectors, i.e., strings of 0s and 1s of some ﬁxed length Û. We deﬁne the operations over bit vectors according their applications to the matching elements of the arguments. For example, we deﬁne Û ½ Û ¾ ¼ & Û ½ Û ¾ ¼ to be Û ½ & & Û ¾ & ¼ , and similarly for operations ˜, |, and ˆ. Letting ¼ ½ Û denot...
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