By Bruck R.H.

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**Extra resources for A survey of binary systems**

**Example text**

Since there is a natural bijection between these numbers and subsets of the set P of all primes in the interval [0, p], the totality ~ becomes a Boolean algebra isomorphic to 2 P • In this situation, the inequality n ~ m means that m is a multiple of n. The product IIq qEP is unity, while the number 1 is the zero element. The role of the supremum for a set of numbers is played by their least common multiple, whereas the infimum is the greatest common divisor. Example 4. 19 By an ~-subspace we mean each I7Cf.

It is introduced by the equality 14 xl y == Cx /\Cy and is remarkable by the fact that the other operations V, /\, and C can be expressed through it. The corresponding formulas will be presented below. We now define the operations of addition and subtraction for elements of a BA. Let E be an arbitrary DISJOINT set. If E possesses a supremum then the latter is called the sum or disjoint sum of E. In this case, we write y=L E xEE rather than y = sup E. y = Xl For finite disjoint sets, we use the notation + X2 + ...

Theorem 2. , X=y implies the other two. 13 Preliminaries on Boolean Algebras Thus, each of the elements 1 x - y 1 and x '" y can be regarded as some MEASURE OF PROXIMITY between x and y. Another binary Boolean operation -+ (implication) is defined by the equality 13 x -+ y == y V Cx. It is easy to verify that the relations x ::; y and x -+ y = II are equivalent. We mention the Sheffer stroke, I. It is introduced by the equality 14 xl y == Cx /\Cy and is remarkable by the fact that the other operations V, /\, and C can be expressed through it.