Boole's expansion theorem by Naftoli Eherenberg, v. 1, ed., page 141) The next day there was another action. The captives this time included three of Halberstam's sons-in-law: Rabbi Yecheskel Halberstam (son of Rabbi Yeshayale Tchechoiver), Rabbi Moshe Stempel, and Rabbi Shlome Rubin. On Monday July 28, the Jewish captives were shot in the Yanover forest behind the city. Twenty thousand Jews were shot that day; Halberstam, his son and his three sons-in-law included. He was succeeded by his son Rabbi Shlomo Halberstam (1907–2000), who rebuilt Bobov in the United States. Ben Zion Halberstam Ben Zion Halberstam (1874–1941) was 	Ben Zion Halberstam
6477386	Aija Province Aija Province The Aija Province is one of 20 provinces of the Ancash Region in Peru. Geography. The Cordillera Negra traverses the province. Some of the highest mountains of the province are listed below: Political division. Aija is divided into five districts, which are: Ethnic groups. The province is inhabited by indigenous citizens of Quechua descent. Spanish is the language which the majority of the population (62.10%) learnt to speak in childhood, while 37.70% of the residents started speaking using the Quechua language (2007 Peru Census). Aija Province The Aija Province is one of 20 provinces of the Ancash Region 	Aija Province
6477387	Boole's expansion theorem Boole's expansion theorem Boole's expansion theorem, often referred to as the Shannon expansion or decomposition, is the identity: formula_1, where formula_2 is any Boolean function, formula_3 is a variable, formula_4 is the complement of formula_3, and formula_6and formula_7 are formula_2 with the argument formula_3 set equal to formula_10 and to formula_11 respectively. The terms formula_6 and formula_7 are sometimes called the positive and negative Shannon cofactors, respectively, of formula_2 with respect to formula_3. These are functions, computed by restrict operator, formula_16 and formula_17 (see valuation (logic) and partial application). It has been called the fundamental theorem of Boolean  algebra". Besides its theoretical importance, it paved the way for binary decision diagrams (BDDs), satisfiability solvers, and many other techniques relevant to computer engineering and formal verification of digital circuits. In such engineering contexts (especially in BDDs), the expansion is interpreted as a if-then-else, with the variable formula_3 being the condition and the cofactors being the branches (formula_6 when formula_3 is true and respectively formula_7 when formula_3 is false). Statement of the theorem. A more explicit way of stating the theorem is: Variations and implications. Repeated application for each argument leads to the Sum of Products (SoP) canonical  form of the Boolean function formula_26. For example for formula_27 that would be Likewise, application of the dual form leads to the Product of Sums (PoS) canonical form (using the distributivity law of formula_29 over formula_30): History. George Boole presented this expansion as his Proposition II, "To expand or develop a function involving any number of logical symbols", in his "Laws of Thought" (1854), and it was "widely applied by Boole and other nineteenth-century logicians". Claude Shannon mentioned this expansion, among other Boolean identities, in a 1949 paper, and showed the switching network interpretations of the identity. In the  literature of computer design and switching theory, the identity is often incorrectly attributed to Shannon. Boole's expansion theorem Boole's expansion theorem, often referred to as the Shannon expansion or decomposition, is the identity: formula_1, where formula_2 is any Boolean function, formula_3 is a variable, formula_4 is the complement of formula_3, and formula_6and formula_7 are formula_2 with the argument formula_3 set equal to formula_10 and to formula_11 respectively. The terms formula_6 and formula_7 are sometimes called the positive and negative Shannon cofactors, respectively, of formula_2 with respect to formula_3. These are functions, computed by restrict operator, formula_16 and formula_17