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Unformatted text preview: haracterized by a set of elements, some of its key operations, and some important elements. As an example, modular arithmetic also forms a ring. For modulus Ò, the algebra is denoted , with components deﬁned as follows: Ò Ò Ò Ò · ¢ ¼½ · Ò Ò Ò ¢ Ò ¼½ ½ · ÑÓ ¢ ÑÓ ¼ ¼ ¼
Ò Ò Ò Ò Even though modular arithmetic yields different results from integer arithmetic, it has many of the same mathematical properties. Other well-known rings include rational and real numbers. End Aside. If we replace the O R operation of Boolean algebra by the E XCLUSIVE -O R operation, and the complement operation ˜ with the identity operation Á —where Á ´ µ for all —we have a structure ¼ ½ ˆ & Á ¼ ½ . This structure is no longer a Boolean algebra—in fact it’s a ring. It can be seen to be a particularly simple form of the ring consisting of all integers ¼ ½ Ò ½ with both addition and multiplication performed modulo Ò. In this case, we have Ò ¾. That is, the Boolean A ND and E XCLUSIVE -O R operations correspon...
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