Chapter 2 Gates

2019/2/27 Binary Logic Deals with binary variables that take 2 discrete values (0 and 1), and with logic operations Three basic logic operations: AND, OR, NOT Binary/logic variables are typically represented as letters: A,B,C,…,X,Y,Z Boolean Algebra 2019/2/27 Boolean Algebra

NOT Gate -- Inverter X Y 1 1

AND Gate AND X Y Z 0 0 0 0 1 0 1 0 0 1 1 1 X Z Y Z = X & Y

OR Gate OR X Y Z 0 0 0 0 1 1 1 0 1 1 1 1 X Z Y Z = X | Y

Basic Logic Gates and Basic Digital Design NOT, AND, and OR Gates NAND and NOR Gates DeMorgan’s Theorem Exclusive-OR (XOR) Gate Multiple-input Gates

NAND Gate NAND X Y Z 0 0 1 0 1 1 1 0 1 1 1 0 X Z Y

NAND Gate NOT-AND X Y W Z 0 0 0 1 0 1 0 1 1 0 0 1 1 1 1 0 X W Z Y

NOR Gate NOR X Y Z 0 0 1 0 1 0 1 0 0 1 1 0 X Z Y

NOR Gate NOT-OR X Y W Z 0 0 0 1 0 1 1 0 1 0 1 0 1 1 1 0 X W Z Y

NAND Gate X X Z Z = Y Y X Y W Z 0 0 0 1 0 1 0 1 1 0 0 1 1 1 1 0 X Y ~X ~Y Z 0 0 1 1 1 0 1 1 0 1 1 0 0 1 1 1 1 0 0 0

NOR Gate X X Z Z Y Y X Y Z X Y ~X ~Y Z 0 0 1 0 0 1 1 1 0 1 0 0 1 1 0 0 0 0 1 0 1 0 1 0 0 1 1 0 X Y ~X ~Y Z 0 0 1 1 1 0 1 1 0 0 1 0 0 1 0 1 1 0 0 0

Exclusive-OR Gate XOR X Y Z X Z 0 0 0 Y 0 1 1 1 0 1 1 1 0

Exclusive-NOR Gate XNOR X Y Z X Z 0 0 1 Y 0 1 0 1 0 0 1 1 1

Multiple-input AND Gate Z 1 Output is HIGH only if all inputs are HIGH Z 1 An open input will float HIGH

Multiple-input OR Gate Z 2 Output is LOW only if all inputs are LOW Z 2

Multiple-input NAND Gate Z 3 Output is LOW only if all inputs are HIGH Z 3

Multiple-input NOR Gate Z 4 Output is HIGH only if all inputs are LOW Z 4

Universal Gates Proving NAND gate is universal

NOR Gate (Universal) Proving NOR gate is universal

Logic Chips (cont.) Integration levels SSI (small scale integration) Introduced in late 1960s 1-10 gates (previous examples) MSI (medium scale integration) 10-100 gates LSI (large scale integration) Introduced in early 1970s 100-10,000 gates VLSI (very large scale integration) Introduced in late 1970s More than 10,000 gates

Logic Design 3-input majority function A B C F 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 1 1 1 1 0 0 0 1 0 1 1 1 1 0 1 1 1 1 1 Logical expression form F = A B + B C + A C