Design Methods for Digital Systems by Jean Chinal (auth.)

By Jean Chinal (auth.)

This e-book constitutes an advent to the idea of binary swap­ ing networks (binary common sense circuits) equivalent to are encountered in business computerized structures, in communications networks and, extra really, in electronic pcs. those common sense circuits, without or with reminiscence, (sequential circuits, combinational circuits) play an expanding half in lots of sectors of in­ dustry. they're, obviously, to be present in electronic desktops the place, through an meeting (often advanced) of elerpentary circuits, the func­ tions of computation and determination that are uncomplicated to the therapy of data, are played. of their flip those pcs shape the center of increasingly more electronic structures to which they're coupled through interface devices which, themselves, fulfil advanced capabilities of knowledge processing. therefore the electronic suggestions penetrate ever extra deeply into business and medical actions within the type of platforms with various levels of specialization, from the wired-in gadget with fastened constitution to these structures situated on a general-purpose programmable com­ puter. moreover, the current chance of mass generating microminiaturi­ sed common sense circuits (integrated circuits, and so forth. ) supplies a foretaste of the intro­ duction of those ideas into the extra time-honored features of lifestyle. the current paintings is dedicated to an exposition of the algebraic techni­ ques nesessary for the research and synthesis of such common sense networks. No past wisdom of this box of task is critical: any technician or engineer owning an straight forward wisdom of arithmetic and electronics can adopt its reading.

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The relation of parallelism between straight lines is thus an equivalence relation. (3) The congruence of integers. Consider the set Z of integers (of either sign). It is said that x and y are congruent with the modulus m (congruent modulo m), when: x - y = km where k is an integer. 30) 0, (- k), kIf are still integers. Thus the congruence relation is an equivalence relation. It is expressed as: x == y (mod. 31) (read as "x congruent with y modulo m", or "x equals y modulo m"). 6 Equivalence Classes For a given element x, the set 0 (x) of elements y which are equivalent to x through the relation R, is known as the equivalence class containing x.

It is said that x and y are congruent with the modulus m (congruent modulo m), when: x - y = km where k is an integer. 30) 0, (- k), kIf are still integers. Thus the congruence relation is an equivalence relation. It is expressed as: x == y (mod. 31) (read as "x congruent with y modulo m", or "x equals y modulo m"). 6 Equivalence Classes For a given element x, the set 0 (x) of elements y which are equivalent to x through the relation R, is known as the equivalence class containing x. o (x) = {y I xRy}.

It follows then that if two equivalence classes are different, they are necessarily disjoint. Finally, every element x E E is squarely placed in an equivalence class since xRx (reflexivity) and thus x E C(x). Rank of an equivalence relation The rank of an equivalence relation R in a set E is the number of equivalence classes into which the relation R partitions the set E. Examples: (1) Consider for example, the set of straight lines in a plane and the relation X / / Y. Each class comprises the set of straight lines parallel with a single direction.

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