HOW DO COMPUTER UNDERSTAND LOGIC

 

HOW DO COMPUTER UNDERSTAND THE BINARY LOGIC?

 

.  I AM GOING TO WRITE ABOUT BINARY LOGIC, THAT MOST OF THE COMPUTER MANUFACTURERS AND DEVELOPERS USE.

                

BINARY LOGIC DEALS WITH VARIABLES THAT TAKE ON TWO DISCRETE VALUES AND WITH OPERATIONS THAT ASSUME LOGICAL MEANING.  THE TWO VALUES THE VARIABLES TAKE MAY BE CALLED BY DIFFERENT NAMES (E.G. TRUE AND FALSE, YES AND NO, ETC.), BUT FOR OUR PURPOSE IT IS CONVENIENT TO THINK IN TERMS OF BITS AND ASSIGN THE VALUES OF 1 AND 0. 

 

BINARY LOGIC IS USED TO DESCRIBE, IN A MATHEMATICAL WAY, THE MANIPULATION AND PROCESSING OF BINARY INFORMATION.  IT IS PARTICULARLY SUITED FOR THE ANALYSIS AND DESIGN OF DIGITAL SYSTEMS.  FOR EXAMPLE, THE DIGITAL LOGICAL CIRCUITS OF MANY CIRCUITS THAT PERFORM BINARY ARITHMETIC ARE CIRCUITS WHOSE BEHAVIOR IS MOST CONVENIENTLY EXPRESSED BY MEANS OF BINARY VARIABLES AND LOGICAL OPERATIONS.  THE BINARY LOGIC TO BE INTRODUCED IN THIS SECTION IS EQUIVALENT TO AN ALGEBRA CALLED BOOLEAN ALGEBRA.

 

BINARY LOGIC CONSISTS OF BINARY VARIABLES AND LOGICAL OPERATIONS.  THE VARIABLES ARE DESIGNATED BY LETTERS OF THE ALPHABET SUCH AS A, B, C, X, Y, Z, ETC., WITH EACH VARIABLE HAVING TWO AND ONLY TWO DISTINCT  VALUES : 0 AND 1.  THERE ARE BASIC LOGIC OPERATIONS: AND, OR AND NOT.

 

·        AND: THIS OPERATION IS REPRESENTED BY A DOT OR BY THE ABSENCE OF AN OPERATOR.  FOR EXAMPLE, X.Y = Z OR XY=Z IS READ “X AND Y IS EQUAL TO Z”.  THE LOGICAL OPERATION  AND INTERPRETED  TO MEAN AND Z = 1 IF AND ONLY IF X = 1 AND Y = 1 OTHERWISE Z = 0. (REMEMBER THAT X, Y AND Z ARE BINARY VARIABLES AND CAN BE EQUAL TO EITHER 1 OR 0 NOTHING ELSE).

·        OR : THIS OPERATION IS SHOWN BY ADDITION SYMBOL.  FOR EXAMPLE,  X + Y = Z IS READ “ X OR  Y IS EQUAL TO Z”      MEANING THAT Z = 1 IF X=1 OR Y=1 OR BOTH X=1 OR IF BOTH X=1 AND Y = 1.  IF BOTH X = 0 , THEN Y = 0 THEN Z = 0.

·        NOT : THIS OPERATION IS PRESENTED BY  A PRIME (SOMETIMES BY A BAR).  FOR EXAMPLE , X’ = Z (OR X NOT EQUAL TO Z MEANING THAT X IS WHAT Z IS NOT) .  IN OTHER WORDS, IF X = 1, AND Z = 0 .  BUT IF X=0 THEN Z = 1.

 

BINARY LOGIC RESEMBLES BINARY ARITHMETIC AND THE OPERATIONS “AND” AND “OR” HAVE SOME SIMILARITIES TO MULTIPLICATION AND ADDITIONS, RESPECTIVELY.  IN FACT, THE SYMBOLS USED FOR AND AND OR ARE THE SAME AS THOSE USED FOR MULTIPLICATION AND ADDITION.  HOWEVER, BINARY LOGIC SHOULD NOT BE CONFUSED WITH BINARY ARITHMETIC.  ONE SHOULD REALIZE THAT AN ARITHMETIC VARIABLE DESIGNATES A NUMBER THAT MAY CONSIST OF MANY DIGITS.  A LOGIC VARIABLE IS EITHER A ONE OR ZERO.  FOR EXAMPLE, IN BINARY ARITHMETIC WE HAVE 1 + 1 = 1 (READ “ONE PLUS ONE EQUAL TO 2” WHILE IN BINARY LOGIC WE HAVE 1 + 1 = 1 (READ “ ONE OR ONE EQUAL TO ONE”

 

FOR EACH COMBINATION OF THE VALUES OF X AND Y THERE IS A VALUE OF Z SPECIFIED BY THE DEFINITION OF THE LOGICAL OPERATION.  THESE DEFINATIONS MAY BE LISTED IN COMPACT FORM USING TRUTH TABLES.  A TRUTH TABLE IS A TABLE OF ALL POSSIBLE COMBINATION OF THE VARIABLES SHOWING THE RELATIONS BETWEEN THE BALUES THAT THE VARIABLES MAY TAKE AND THE RESULT OF THE OPERATION.  FOR EXAMPLE, THE TRUTH TABLES FOR HE OPERATIONS AND AND OR WITH VARIABLES X AND Y ARE OBTAINED BY LISTING ALL POSSIBLE VALUES THAT THE VARIABLE MAY HAVE WHEN COMBINED IN PAIRS.  THE RESULT OF THE OPERATION FOR EACH COMBINATION IS WHEN LISTED IN A SEPARATE ROW.  THE TRUTH TABLES FOR “AND” , “OR” AND “NOT” ARE AS UNDER.

 

 

                                    AND                        

X	y	x.y
0	0	0
0	1	0
1	0	0
1	1	1

 

                                     OR

 

X	y	x +  y
0	0	0
0	1	1
1	0	1
1	1	1

 

 

                                  

 

                      NOT

X	x’
0	1
0	1
1	0
1	0

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