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ME430 Mechatronic Systems Boolean Logic, Truth Tables, and Gate Diagrams
Boolean Logic: Boolean Logic is named after George Boole who was an English Mathematician who constructed the logical algebra we're about to look at.
-- he was the son of a shoemaker and his formal education ended in the 3rd^ grade. -- he produced significant contributions in the areas of differential and difference equations as well as algebra.
The basis of Boolean logic consists of three Logical operations: Logical Addition (which we will call Logical OR ) A + B Logical Multiplication (which we will call Logical AND ) A · B Logical complementation (which we will call Logical NOT )
Logical OR Logical AND Logical NOT
A
Table 1: Definitions for discrete Logic
True False
Logic True = 1 Logic False = 0 In Circuits True is often = 5 volts In Circuits False is = 0 volts High Low On Off
A B A+B A B A · B A
A
Hierarchy of Operations: When presented without parenthesis to indicate which operation to perform first, use the following order for importance:
Most importance: NOT AND Least importance: OR
Example: Complete the truth table for the Logical Expression:
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X Y Z
X+Y Z⋅
Y Y Z⋅ X+(Y Z)⋅ X+Y Z⋅
DeMorgan Theorem are statements that show relationships between NAND and NOR gate structures.
and
NOR NAND
Examine the Truth Table for each form of the NOR:
A similar relationship is also true for the NAND gate. Examine the Truth Table for each form of the NOR:
A NOR may be created by either A NAND may be created by either -- inverting the output of an OR gate --inverting the output of an AND gate or or -- inverting all inputs of an AND gate -- inverting all inputs of an OR gate.
A B
A B
( X+Y =X Y) ⋅^ ( X Y =X+Y⋅ )
A+B
A B A B⋅
A B
A B⋅ A B A+B
Fill in the final column for each truth table about and try to figure out what the results of the truth tables tell you about the NAND gate? Does the final column look familiar? Other gates?
If you look at the final columns you can see that any other kind of logic gate may be set up using only NAND (in the right number and configuration). The first truth table is the same as a NOT gate, the seond is the same as an AND gate, the third is an OR gate. So you can make NOT, AND, or OR using nothing but NAND gates. You don’t even need the other types of gates. You can do everything with NANDs if you wanted.
Sometime it’s not the most efficient system to use only NAND gates (the above is NOT efficient). But sometimes it’s really useful to use only one gate type. So if you ever feel the urge you can make any discrete logic circuit using only NAND gates.
A
A B
A B
A A⋅
A B⋅ A B⋅ ⋅ A B⋅
A A⋅ B B⋅ A A⋅ ⋅ B B⋅
A (^)?
A
B^?
B
A ?
NANDs and NORs have a unique and robust ability. If you could only have a single kind of gate, you'd probably want to have either all NANDs or all NORs. To understand why, determine the logical results of the circuits below.
Example of data sheet for a quad 2 input NAND gate.
….more of the data sheet…..
Solutions from earlier in the handout
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Solution to Problem 1
X Y Z 0 0 0 1 0 0 1 0 0 1 1 1 1 0 0 1 0 0 0 0 1 0 1 1 0 0 0 1 1 0 0 1 0 1 0 1 0 1 1 1 1 0 1 1 0 0 0 1 0 1 1 1 0 0 1 0
Y Y Z⋅ X+(Y Z)⋅ X+Y Z⋅
