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Additional gates We’ve already seen all the basic Boolean operations and the associated primitive logic gates. There are a few additional gates that are often used in logic design. They are all equivalent to some combination of primitive gates. But they have some interesting properties in their own right.

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Additional Boolean operations x y (xy)’ 0 0 1 0 1 1 1 0 1 1 1 0 x y (x+y)’ 0 0 1 0 1 0 1 0 0 1 1 0 NAND (NOT-AND) NOR (NOT-OR) XOR (eXclusive OR) (xy)’ = x’ + y’ (x + y)’ = x’ y’ x y = x’y + xy’ Operation: Expressions: Truth table: Logic gates: x y x y 0 0 0 0 1 1 1 0 1 1 1 0
NANDs are special! The NAND gate is universal : it can replace all other gates! NOT AND OR (xx)’ = x’ [ because xx = x ] ((xy)’ )’ = xy [because (x’)’ = x] ((xx)’ (yy)’)’ = (x’ y’)’ [ xx = x, and yy = y ] = x + y [ DeMorgan’s law ]

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Making NAND circuits The easiest way to make a NAND circuit is to start with a regular, primitive gate-based diagram. Two-level circuits are trivial to convert, so here is a slightly more complex random example.
Converting to a NAND circuit Step 1: Convert all AND gates to NAND gates and convert all OR gates to NAND gates. AND AND AND AND OR OR

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Converting to NAND, concluded Step 2: Cancel all pairs of inverters ((x’)’ = x). .
The NOR gate is also universal . The NOR gate is the dual of the NAND NOT AND’ OR NORS are also special! (x+x)’ = x’ ((x+y)’ )’= x+y (x’ + y’)’ = (x’)’ (y’)’ = x y

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NOR gates The NOR operation is the dual of the NAND. NOR gates are also universal. We can convert arbitrary circuits to NOR diagrams by following a procedure similar to the one just shown: Step 1: Convert all OR gates to NOR gates and all AND gates to NOR gates. Step 2: Cancel pairs of inverters
A two-input XOR gate outputs true when exactly one of its inputs is true: XOR corresponds more closely to typical English usage of “or,” as in “eat your vegetables or you won’t get any pudding.” Several fascinating properties of the XOR operation: x y x y 0 0 0 0 1 1 1 0 1 1 1 0 x y = x’ y + x y’ x 0 = x

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