Logic
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NOT logic gate
▶o
AND logic gate
D
OR logic gate
)> (concave and pointy ver of and)
XOR logic gate
) )> (same as or with added curved line in front)
AND boolean symbol
OR boolean symbol
∨
XOR boolean symbol
v with line underneath
NOT boolean symbol
¬
order of precedence for boolean expressions
Brackets NOT AND OR and XOR
Double Negation
If there are two nots in front of a letter they can be erased ¬¬A = A
Commutation
The order of letters around an operator can be changed (remember the NOT symbol stays with the letter) A v ¬B = ¬B v A
Association
If a long expression contains only one operator, letters can be reordered, and brackets can be added or taken away A ^ B ^ ¬C = ( A ^ ¬C) ^ B
Distribution
If there is more than one operator, letters outside brackets can be “distibruted” to the letters inside the brackets A v ( B ^ C ) = ( A v B) ^ ( A v C )
De Morgan’s Law
If there is a NOT outside of a bracket, it can be added to the letters inside, but the operator must change ¬( A v B) = ¬A ^ ¬B
What is a Karnaugh Map
• A summary version of a truth table used to simplify logic expressions • Can have up to 4 inputs (2 per axis)
How are inputs written on the axis of a Karnaugh map
not in binary number order. They are written like this : 00 01 11 10 so that only 1 bit changes at a time.
How Karnaugh maps correspond to a truth table
Each row of a truth table matches a square of a karnaugh map
How Karnaugh maps correspond to a logical expression
• look at each part of the expression • find the value of that part (such as AvB) and fill in a 1 in all squares that match that condition.
How to mark groups in a karnaugh map
• Must have 1, 2, 4, or 8 1s. • Squares or rectangles only • Can overlap • no 1s must be left out • groups as big as possible
How to use karnaugh map to create logical expression
• Describe each group, connect using AND operator.