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Boolean Algebra
Published in Sajjan G. Shiva, Introduction to Logic Design, 2018
A set of operators is said to be functionally complete if any Boolean function can be expressed in terms of the set. Thus, the set containing AND, OR, and NOT is functionally complete. If it can be shown that a set of operators can realize AND, OR, and NOT operations, then that set is also functionally complete. It can be shown that NAND and NOR are each functionally complete. Figure 2.11 shows how the three basic operators are realized by NAND gates only. Proving that NOR is functionally complete will be left as an exercise. NAND and NOR are also called universal operators. The advantage of universal operators is that only one type of gate is used for realizing the logic circuit. Since integrated circuit (IC) chips contain several gates of the same type, using universal gates in the circuit is advantageous in terms of minimizing the number of chips used in the design. We will examine this further in subsequent chapters.
Programmable Logic Devices
Published in Joseph Cavanagh, Digital Design and Verilog HDL Fundamentals, 2017
As can be seen from the above examples, the output function may not use all of the input variables xi. Combinational logic networks can be characterized in terms of fun-damental logic operations such as, AND, OR, and NOT. This also includes the functionally complete set of logic gates, NAND and NOR. The output symbols zi may be asserted either high or low. Combinational logic is used extensively to represent the next-state function and the output function for both synchronous and asynchronous sequential machines.
Boolean Algebra and Implementation
Published in Eugene D. Fabricius, Modern Digital Design and Switching Theory, 2017
NAND gates are functionally complete of themselves, as are NOR gates, since both can invert signals without additional help. The realization of all three boolean operations using only NAND gates, and the realization of all three boolean operations using only NOR gates is shown in Figure 2.23.
A note on the complexity of S4.2
Published in Journal of Applied Non-Classical Logics, 2021
Aggeliki Chalki, Costas D. Koutras, Yorgos Zikos
In many cases below, we shall tacitly assume a functionally complete set of propositional/boolean connectives (e.g. , ) in our language, to simplify proofs by structural induction. Also, we will make use of the abbreviations and .