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Concise text begins with overview of elementary mathematical concepts and outlines theory of Boolean algebras; defines operators for elimination, division, and expansion; covers syllogistic reasoning, solution of Boolean equations, As a matter of fact, in those early days the main interest was concentrated on what we call today totally ordered sets or chains, that is, sets endowed with a relation < which satisfies (5)–(7) and (9) x = y =⇒ x<y or y<x. Besides, in the years 18901930 the abstract relation < was not yet well crystallized and many authors worked with <, > and = as three interconnected primitive concepts. So did Betazzi [1890]*, BuraliForti [1893]*, Stoltz and Gmeiner [1901]*, Shatunovskiˇi [1904]*, For.the fourth relation : F(x) F(y) = F(x n Qy) + F(x n y) (F(y n f» + F(x n y)) <, ^F(xnCy) +F(y nCx), etc. 4.5.9 Corollary, x > £x, (x,y)>xUy, {x,y)+xny and F are uniformly continuous with respect to pF. 4.5.10 Proposition. If £2 is a boolean algebra, JFeZ(Q)*, F > 0 F is faithful and ( Q, pF) is a complete metric space then £2 is a boolean aalgebra and F e * ( £}). Proof. Let xn e Q and consider y„ = xxU U xn. F(y„) is an ascending sequence hence it is convergent, i.e. pF(y„, ym)= \F(yn) But then there is a term m with (x,m(x,y, z)) 6 9(x, z) A 9(x,y) and (m(x,y, z),z) £ 9(x, z) A 9(y, z), and this term m satisfies the identities needed to make it a majority term. (ii) => (iii): Given terms.p and m as in (ii), we take q(x,y,z) to be the term p(x,m(x, y, z), z). It can be shown that the variety of all Boolean algebras is generated by the twoelement Boolean algebra 2g — ({0, 1}; A, V, >,=>, 0, 1), where the operation symbols here denote the usual Boolean operations on the set {0,1}.iA(^Xy)^Xy. For (* X y) X 1 = * X y. T4.14. xXx = x=>x = 0orx= 1. With the aid of these theorems it is not difficult to verify that a system B satisfying the above nine postulates is a Boolean algebra provided /\ v are defined by D4.1, D4.3 and ~# is denned to be 1 + x. Moreover, the operator C defined by D4.2 satisfies postulates P3.1 to P3.7. From T4.13, T4.14 it follows that a nontrivial implicative Boolean algebra is atomless. In.fact every element is infinitely divisible. Since there exist atomic 9.2. DEFINITION: BOOLEAN. ALGEBRA. A nonempty set B consisting of at least two elements 0 and 1 and two binary operations (+) and called addition and multiplication and a unitary operator called complementation (denoted by ′) is called a Boolean algebra provided that the operations satisfy the following axioms for all x, y, and z in B. 1 As proven in the general theory of uniform spaces, each uniformity can be determined from a set of pseudometrics.11 We now establish this fact in a form adapted to Boolean algebras. {Vn} satisfying v, C V. A finite real function 9 on 3£ is called an outer quasimeasure whenever 9 satisfies the following conditions for all x,.y C 3£: = 0; (2) if x < y then 9(x) <0(y); We call a system M of outer quasimeasures sufficient whenever 9Jt possesses the properties: a) supe€mO(x) > 0 for all x > 0, 3+4 Simplify the Boolean expression F(A,B,C,D) = X(0, 1, 2, 3, 5, 8, 10, 11), dX(4, 7, 9), using K'map. Convert the expression F(X,Y,Z) = X(Y+Z)(X+Z) into canonical POS form. 3+4 Write the names of the universal gates. Draw the truth table of an XNOR gate. Simplify the Boolean expression F(A,B,C,D) = TI(1, 2, 8, 14) using K'map. 2+2+3 Draw the circuit diagram for the SOP expression f = A B+ AB + AB. Show how to form an OR logic gate using only NAND gates. 4+3 Draw the circuit We review some of the basic relationships in Boolean algebra which.are essential in the simplification of fault trees. More comprehensive rules of Boolean algebra are available in [4] or any of the many textbooks on the subject. Let us consider a pair of variables X and Y and explore the rules of Boolean algebra and their equivalent engineering symbolism. Let X be the set {2, 4, 6, 8} and Y be the set {1, 3, 6, 8, 9, 10} 1. X\Y: The set of elements which belong to X AND Y. This is also