Union, Boolean

Boolean multiplication is distributive with respect to Boolean union that is,... 

Boolean union, on the other hand, differs significantly from addition. We have, for example,... 

One method of locating these maximal loops is to compute the reachability matrix, R (H1), which is the element by element Boolean union of all of the powers of the adjacency matrix up to the nth, where n is the number of rows of R. An element of the reachability matrix is defined as... 

One method of partitioning the system equations is to compute the maximal loops using powers of the adjacency matrix as discussed in Section II. Certain modifications to the methods of Section II are needed in order to reduce the computation time. The first modification is to obtain the product of the matrices using Boolean unions of rows instead of the multiplication technique previously demonstrated to obtain a power of an adjacency matrix. To show how the Boolean union of rows can replace the standard matrix multiplication, consider the definition of Boolean matrix multiplication, Eq. (2), which can be expanded to... 

This modification for Boolean matrix multiplication permits use of the Boolean union operation (logical OR operation or logical sum) instead of regular multiplication and union operations. The Boolean union operation can be executed much faster on a digital computer. Experience has shown that performing the Boolean union of rows instead of the standard Boolean multiplication of matrices can reduce the computation time by as much as a factor of four. 

There are no rows with all zero elements in the reduced matrix of Fig. 9a, hence we proceed to the second phase of Steward s algorithm by starting with the first row of the reduced matrix to trace the path 2 - 3 -> 4 -> 2. The loop of information flow between vertices 2, 3, and 4 is encircled by the dashed line in the graph in Fig. 9b. The rows and columns labeled 2, 5, and 4 are next removed from the reduced matrix and one row, which is the Boolean union of the rows labeled 2, 3, and 4, and one column, which is the Boolean union of the columns labeled 2, 2, and 4, are added to the reduced matrix to obtain the new reduced matrix of Fig. 10a. The added row and column are labeled... 

Form a new j by n Boolean matrix, M(0) as follows For each zero entry in column k, reproduce the corresponding row as a row in M(0). For example, the second row of the occurrence matrix in Fig. 12a contains a zero in column k = 1 and therefore the element = 0, element — 1, to 3 = 0, and to 4 = 1 comprise the first row of M(0). The second row of M(0) would be exactly like the third row of that occurrence matrix. This process is continued until all of the rows with zero entries in column k have been included as rows of M(0). A final row is added to M(0), whch is the Boolean union of all of the rows in the occurrence matrix which contain nonzero entries in column k. For example, rows 1 amd 4 of the occurrence matrix contain nonzero elements in column 1 so that the elements of the last row of M(0) are m3I = 1, m32 = 0, m33 = 1, m34 = 0. Figure 13a illustrates M(0). 

A new matrix M(1) is formed from M(0) in the same manner that M(0) was formed from the occurrence matrix. The column of M<0) containing the most nonzero elements is identified, and M(1> is made up of rows identical to each row of M(0), which contains a zero in column k and one final row, which is the Boolean union of the remaining rows in M(0). Figure 13b illustrates M(1). A record is kept of which rows of the original occurrence matrix have been combined to form each row of M(0), M(1), and so on. 

C IF THE VARIABLE CAN BE AN ITERATE THE BOOLEAN UNION OF THE COLUMN C ANO THE LAST LOOP COLUMN IS PLACED IN THE LAST LOOP COLUMN. 

I Figure 16iS Examples of Boolean union (add) and subtract operations. 

Union The union of two fuzzy sets A and B eorresponds to the Boolean OR funetion and is given by... 

Table 2.1-1 compares the ordinary algebra of continuous variables with the Boolean algebra of 1 s and Os. This table uses the symbols and -h for the operations of intersection (AND) and union (OR) which mathematicians represent by n and u respectively. The symbols and -f which are the symbols of multiplication and addition, are used because of the similarity of their use to AND and OR in logic. 

Exercise Draw diagrams to illustrate the notions of fcontainment, union, and intersection, and the properties of a Boolean algebra. Hence, a and b are independent if, and only if... 

Any fuzzy power set with the subsethood relation is a lattice, in which the standard fuzzy intersection and union play the roles of the meet and the join, respectively. The lattice is distributive and complemented under the standard fuzzy complement. Contrary to the Boolean lattice, which is associated with classical power sets, it does not satisfy the law of the excluded middle and the law of contradiction. Such a lattice is usually called a DeMorgan lattice. 

In the cases of pericyclic reactions with an odd number of centers and condensed aromatic compounds, we obtain a poset (L. v) with respect to the union of edge sets. Its underlying abstract diagram is isom< phic to the diagram of a Boolean lattice, but its... 

The results of a Select statement are in the form of a table. This can be a subset of a single table, or the result of joining several tables. The exact set of rows is chosen by using various Where clauses. The use of Boolean operation such as and, or, and not allows a sort of union (or), intersection (and), and difference (not). For example ... 

Ri ardless of how you generate areas or volumes, you can use Boolean operations to add (union) or subtract entities to create a solid modd. Examples of Boolean operations are shovm in Figure 16.25. 

Here the-E (Boolean OR) symbol represents the union symbol U, and the symbol (Boolean AND) represents the intersection symbol n.)... 

These relationships can be developed and dealt with according to the rules of Boolean algebra, which are similar yet not identical to those of the ordinary algebra. Some of these rules and properties are listed in Table 11-2 (it must be remembered that the -E (OR) symbol and the (AND) symbol mean union and intersection , respectively). 

SMP, just as Smax, associates with each object the list of its main characteristics value type, dimensions, initial value, transfer function, and whether or not it is memorizing. If the object is a vector, there may be different transfer functions associated to distinct slices of the vector. The general form of a transfer function is a set of conditional transfers, where all conditions must be mutually exclusive, and the union of all conditions is equal to 1 for non memorising objects. All the classical boolean functions are pre-defined in SMP, on scalars as well as vectors, and new functions are constructed by composition of existing ones, in standard prefix notation. SMP is coded as Common LISP property lists, and in this sense is more abstract than Blif-mv. 

Technically, if v(S) w(S) 0 implies t if . In general, the lattice property refers to the existence, for any two elements v and w, of a greatest lower bound V niW and a least upper bound v U w in Above, = v U 0 and v = v C 0. Unions and intersections define Boolean algebras as lattices with special additional properties. For the facts about Banach lattices needed here, see Garrett Birkhoff, Lattice theory, rev. ed.. New York, 1948, Chapter XV. 

An alternative way to think of Definition 9.14 is the following. Assume that we have n data points and N landmark points. Every data point induces an order on the landmark points just sort them with respect to their distances to that point. Every such ordering can be visualized as a path in the Hasse diagram of the Boolean lattice starting from the point nearest to the chosen data point, then proceeding to the union of the two closest ones, then on to the three closest ones, and so on. Now, the witness complex W(A, B) is the maximal abstract sirnplicial complex whose face poset is contained in the union of these paths. 

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