Logic Gate Level. Combinational Circuits, Part 1. Circuits. Circuit: collection of devices physically connected by wires to form a network Net can be: combinational: input determines output sequential: output not entirely dependent on input We will focus first on combinational networks.
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Combinational Circuits, Part 1
Black box view of network
Inputs: a, b, c
Outputs: x, y
Note, in particular, how the distributive property of Boolean algebra differs from regular algebra – in regular algebra, multiplication (denoted with ) distibrutes over addition (+), but not vice versa, as the operations represented by these symbols (AND and OR) do in Boolean algebra
x + x = x
x + x = (x + x) 1 by the identity property
(x + x) 1 = (x + x) (x + x’) by complement
(x + x) (x + x’) = x + (x x’) by distribution
x + (x x’) = x + 0 by complement
x + 0 = x by identity
a b a b
1 1 0
1 0 1
0 1 1
0 0 0
NAND and NOR gates are logically equivalent to AND and OR gates followed by inverters. Since NAND is actually easier to build, ANDs are often constructed as inverted NAND gates.