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CONSTRUCTING CIRCUITS USING BINARY LOGIC GATES - Coggle Diagram
CONSTRUCTING CIRCUITS USING BINARY LOGIC GATES
BOOLEAN OPERATION AND EXPRESSIONS
BOOLEAN EXPRESSION ARE USED TO GUIDE IN BUILDING LOGIC CIRCUITS
THE BOOLEAN EXPRESSIONS CAKKED BOOLEAN VARIABLES
the AND GATE operation
the output of an AND GATE is HIGH only when all inputs are HIGH
the OR GATE operation
the output of an OR GATE is HIGH whenever one or more inputs are HIGH
the NOT GATE operation
the output of an inverter is always the complement (opposite) of the input
BOOLEAN VARIABLES
BOOLEAN VARIABLES TAKE THE VALUE EITHER 0 or 1 ONLY
IN DIGITAL ELECTRONICS
BOOLEAN 0 and 1 CORRESPOND TO THE BINARY 0 and 1
IN LOGIC
1 and 0 ARE SOMETIMES CALLED TRUE AND FALSE
WE USE SYMBOLS RO REPRESENT BOOLEAN VARIABLES
JUST LIKE WITH ORDINARY ALGEBRA
TYPICALLY A SINGLE CHARACTER
TYPICALLY UPPER CASE
DESCRIBING LOGIC CIRCUIT USING BOOLEAN EXPRESSION
ANY LOGIC CIRCUIT , NO MATTER HOW COMPLEX , CAN BE COMPLETELY DESCRIBED USING THE THREE BASIC BOOLEAN OPERATIONS ; OR,AND , NOT
LAW AND RULES BOOLEAN ALGEBRA
BOOLEAN ALGEBRA IS A DIGITAL ELECTRONIC SYSTEMS MANIPULATE BINARY INFORMATION
TO DESIGN SUCH SYSTEMS WE NEED A CONVENIENT MATHEMATICAL FRAMEWORK
BOOLEAN ALGEBRA PROVIDES THIS FRAMEWORK
POINTS IN A CIRCUIT ARE REPRESENTED BY BOOLEAN VARIABLES
BOOLEAN ALGEBRA ALLOWS US TO SPECIFY RELATIONSHIPS BTW BOOLEAN VARIABLES
CAN BE USED AS A DESIGN TOOL FOR DIGITAL ELECTRONIC CIRCUITS
FOR 2 BINARY VARIABLES (TAKING VALUES 0 and 1) THERE ARE 16 POSSIBLE FUNCTIONS
INVOLVE ONLY THREE OPERATIONS WHICH MAKE UP BOOLEAN ALGEBRA: AND ,OR, and COMPLEMENT
THESE OPERATIONS ARE LIKE ORDINARY ALGEBRAIC OPERATIONS IN THAT THEY ARE COMMUTATIVE,ASSOCIATIVE AND DISTRIBUTIVE.
THERE IS A GROUP OF USEFUL THEOREMS OF BOOLEAN ALGEBRA WHICH HELP IN DEVELOPING THE LOGIC FOR A GIVEN OPERATION.
BOOLEAN ALGEBRA THEOREM
AND,OR are associative
AND and OR operations are commutative
forms of the distributive property
a form of DeMorgan's Theorem
THE MOST IMPORTANT LOGIC THEOREM FOR DIGITAL ELECTRONICS, THIS THEOREM SAYS THAT ANY LOGICAL BINARY EXPRESSION REMAINS UNCHANGED IF WE :-
CHANGE ALL VARIABLES TO THEIR COMPLEMENTS
change all AND operations to ORs
change all OR operations to ANDs
TAKE THE COMPLEMENT OF THE ENTIRE EXPRESSION
TWO FORMS OF DeMorgan's Theorem implemented with basic gates
Single Variable Theorems
More two-variable Theorems
Identity and Null operations
ALGEBRAIC PROPERTIES OF REAL NUMBERS
COMMUTATIVE
ASSOCIATIVE
DISTRIBUTIVE
SOP Karnaugh Map (K-map)
USE TO SIMPLIFY BOOLEAN EXPRESSIONS ALSO KNOW AS K-MAP
TRANSFERRING NORMAL TRUTH TABLE FOR LOGIC CIRCUIT INTO A TABLE CALLED K-MAP TABLE
SUITABLE FOR SINGLE OUTPUT WITH THREE TO SIX INPUT
THIS METHOD WILL DISPLAY THE RELATIONSHIP BTW bit MORE EFFECTIVE
STANDARD TRUTH TABLE-a SINGLE DIMENSIONAL PRESENTATION OF A LOGIC
THIS METHOD IS FAIRLY LABOR INTENSIVE AND WILL BURN UP A LOT OF PAPER
ALLOWS FOR QUICK SCANNING OF ACTIVE OUTPUT bits AGAINST THEIR INPUTS,TO FIND BASIC RELATIONSHIPS BTW THEM
TRANSFER TRUTH TABLE INTO K-MAP
LOOPING
TO SIMPLIFY BOOLEAN EXPRESSION
START WITH MINTERM BOOLEAN EXPRESSION
RECORD 1s ON A KARNAUGH MAP
LOOP ADJACENT 1s (LOOPS OF TWO,FOUR,EIGHT SQUARES)
SIMPLIFY BY DROOPING TERMS THAT CONTAIN A TERM AND ITS COMPLEMENT WITHIN A LOOP
OR the remaining terms (one term per loop)
WRITE THE SIMPLIFIED MINTERM BOOLEN EXPRESSION
KARNAUGH MAP OF MAXTERM (POS)
A MINTERM OR SUM-OF-PRODUCT (SOP) IS A BOOLEAN EXPRESSION RESULTING IN 1 FOR THE OUTPUT OF A SINGLE CELL AND 0s FOR ALL OTHER CELLS IN THE KARNAUGH MAP OR TRUTH TABLE
A BOOLEAN EXPRESSION RESULTING IN A 0 FOR THE OUTPUT OF A SINGLE CELL AND 1s FOR ALL OTHER CELLS IN THE KARNAUGH MAP OR TRUTH TABLE
PROCEDURE FOR PLACING A MAXTERM IN THE K-MAP
IDENTIFY THE SUM TERM TO BE MAPPED
WRITE CORRESPONDING BINARY NUMERIC VALUE
FORM THE COMPLEMENT
USE THE COMPLEMENT AS AN ADDRESS TO PLACE A 0 IN THE K-MAP.
REPEAT FOR OTHER MAXTERMS