[home] [lexicon] [problems] [tests] [courses] [auxiliaries] [notes] [staff]  
MathematicsOnline course: Basic Mathematics  Propositional Logic  
Logical Operations 
[previous page] [next page]  [table of contents][page overview] 
Operation  Notation  (read as)  is true if and only if  
Negation  (not )  is false  
Conjunction  ( and )  both and are true  
Disjunction  ( or )  or is true (or both are true)  
Antivalence  (either or )  and are assigned different truth values  
Implication 

is false or is true  
Equivalence  ( is equivalent to )  and are assigned identical truth values 
In order to reduce the usage of parentheses in logical formulas, we define that is more closely linked to a symbol than and , which in turn are more closely linked than , and .
Note that an implication
only requires the truth of if is true.
A false proposition implies anything, hence both true and false implications can be drawn.
Usually, the orconnective is symbolised by a v, derived from the word vel (Latin: or), yet it is also
common practice
to use the symbol ,,``; then ,,``symbolizes the andconnective. Using 0 to refer to the truth value ,,false``while interpreting any other value as ,,true``enables us to determine the truth value of logical formulas via calculation with natural numbers.
Particularly computerlinguists frequently use the English terms NOT (negation), AND (conjunction), OR (disjunction), EXOR or XOR (exclusive or, antivalence) as well as their negations NAND (negated conjunction), NOR (negated disjunction) and NXOR (equivalence).
andconnective  orconnective 
A negated statement corresponds to a switch that is closed if the statement is false. Thus it is possible to draw circuits representing equivalence, antivalence and implication.
Equivalence: which can be rewritten as 
Antivalence: which can be rewritten as 
Implication: which can be rewritten as 
Switches can be represented by transistors, for example, that conduct electricity when a high or low voltage is impressed. Values w and f (or 1 and 0) represent high and low voltages respectively.
DIN 40900 defines symbols for the corresponding circuits. These consist of rectangles in which the respective operations are inscribed. Negation is symbolized by a circle.
Conjunction  Disjunction  Antivalence 
Negation  Implication  Equivalence 
[previous page] [next page]  [table of contents][page overview] 
automatically generated 10/31/2008 