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Unformatted text preview: c A.H.Dixon CMPT 150: Week 5 (Feb 4 - 8, 2008) 31 This means NAND can be used to replace the AND and OR gates in a cir- cuit: C ( x,y,z ) = xy + xz + yz First, observe that • x ↑ y = ( xy ) . Therefore an AND gate whose output is inverted can be replaced by a NAND gate. • x ↑ y = ( x y ) = x 00 + y 00 = x + y . Therefore an OR gate, can be replaced by a NAND gate with complemented inputs. • x ↑ x = ( xx ) = x (by the Idempotent Law). Thus by connecting the two inputs of a NAND gate together, one obtains a NOT gate. • x 00 = x . (This is the Involution Law). Therefore two Inverters in series is logically equivalent to an ungated signal line. One can introduce two inverters in a row without changing the logical behavior of the circuit. To obtain a circuit that uses only NAND gates from a circuit that uses AND, OR, and NOT gates, replace each AND, OR, and NOT gate by a ”subcircuit” that implements the AND, OR, or NOT gate using only NAND gates. That is: xy + xz + yz = (( xy ) ( xz ) ( yz ) ) = ( xy ) ↑ ( xz ) ↑ ( yz ) ( xy ) = ( x ↑ y ) ( xz ) = ( x ↑ z ) ( yz ) = ( y ↑ z ) C ( x,y,z ) = ( x ↑ y ) ↑ ( x ↑ z ) ↑ ( y ↑ z ) Notice that when the sum-of-products representation of a circuit is used, then a substitution of NAND gates for AND and OR gates can be made on a ”one-for- one” basis. Now since every circuit can be implemented as a sum-of-products, it follows that every circuit can be implemented using only NAND gates following the procedure suggested by the example. c A.H.Dixon CMPT 150: Week 5 (Feb 4 - 8, 2008) 32 13.2 NOR Substitution NOR can be used to replace the AND and OR gates in a circuit. Product-of-sums representations can be implemented using a single type of gate, the NOR gate, since: • x ↓ y ) = ( x + y ) Therefore an OR gate whose output is inverted can be replaced by a NOR gate. • x ↓ y = ( x + y ) = x y Therefore an AND gate, all of whose inputs are complemented can be replaced by a NOR gate. • x ↓ x = ( x + x ) = x Therefore a NOR gate can be converted to a NOT GATE by ”tying its inputs together”. Compare these observations with the example above using NAND gates and note how the duality principle has been applied. In the case of NOR, the starting point is a product of sums, rather than a sum of products, since with such representations all OR and AND gates can be replaced by NOR gates on a one-for-one basis....
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This note was uploaded on 06/20/2010 for the course CMPT 150 taught by Professor Dr.anthonydixon during the Spring '08 term at Simon Fraser.
- Spring '08