2's Complement Circuit

This article introduces various Boolean operators which are used in discussing the properties and stability of a 2's complement circuit. We present the deenitions and related theorems for the following logical operators which include negative input/output: 'and2a', 'or2a', 'xor2a' and 'nand2a', 'nor2a', etc. We formalize the concept of a 2's complement circuit, deene the structures of comple-mentors/incrementors for binary operations, and prove the stability of the circuit. provide the notation and terminology for this paper. Let x be a set. Then hxi is a nite sequence with length 1. Let y be a set. Then hx; yi is a nite sequence with length 2. Let z be a set. Then hx; y; zi is a nite sequence with length 3. Let n, m be natural numbers, let p be a nite sequence with length n, and let q be a nite sequence with length m. Then p a q is a nite sequence with length n + m: Let S be an unsplit non void non empty many sorted signature, let A be a Boolean circuit of S, let s be a state of A, and let v be a vertex of S. Then s(v) is an element of Boolean. We now state two propositions: (1) For every function f and for all sets x 1 , x 2 such that x 1 2 dom f and x 2 2 dom f holds f hx 1 ; x 2 i = hf (x 1); f(x 2)i: (2) For every function f and for all sets x 1 , x 2 , x 3 such that x 1 2 dom f and x 2 2 dom f and x 3 2 dom f holds f hx 1 ; x 2 ; x 3 i = hf (x 1); f(x 2); f(x 3)i: The function and 2 from Boolean 2 into Boolean is deened by: (Def. 1) For all elements x, y of Boolean holds and 2 (hx; yi) = x ^ y: The function and 2a from Boolean 2 into Boolean is deened as follows: (Def. 2) For all elements x, y of Boolean holds (and 2a)(hx; yi) = :x ^ y: The function and 2b from Boolean 2 into Boolean is deened by: (Def. 3) For all elements x, y of Boolean holds (and 2b)(hx; yi) = :x ^ :y: