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Mathematics-Online course: Basic Mathematics - Propositional Logic

Logical Operations

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Logical statements may be joined via operations listed in the following table.

Operation Notation (read as) is true if and only if
Negation $ \lnot A$ (not $ A$) $ A$ is false
Conjunction $ A\land B$ ($ A$ and $ B$) both $ A$ and $ B$ are true
Disjunction $ A\lor B$ ($ A$ or $ B$) $ A$ or $ B$ is true (or both are true)
Antivalence $ A \not\equiv B$ (either $ A$ or $ B$) $ A$ and $ B$ are assigned different truth values
$ A\Rightarrow B$  
$ B\Leftarrow A$  
($ A$ implies $ B$)  
($ B$ follows from $ A$)  
$ A$ is false or $ B$ is true
Equivalence $ A\Leftrightarrow B$ ($ A$ is equivalent to $ B$) $ A$ and $ B$ are assigned identical truth values

In order to reduce the usage of parentheses in logical formulas, we define that $ \lnot$ is more closely linked to a symbol than $ \land$ and $ \lor$, which in turn are more closely linked than $ \Rightarrow$, $ \Leftrightarrow$ and $ \not\equiv$.

Note that an implication $ A\Rightarrow B$ only requires the truth of $ B$ if $ A$ is true. A false proposition implies anything, hence both true and false implications can be drawn.

Usually, the or-connective is symbolised by a v, derived from the word vel (Latin: or), yet it is also common practice to use the symbol ,,$ +$``; then ,,$ \cdot$``symbolizes the and-connective. 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 computer-linguists 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).

(Authors: Höllig/Hörner/Kimmerle/Abele)

Displaying statements as switches that are closed if the statement is true (and open if the statement is false, respectively), the and-connective can be represented by a serial connection and the or-connective by a parallel connection.
and-connective or-connective
\includegraphics{logische_schaltungen_und} \includegraphics[origin=tl]{logische_schaltungen_oder}

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: $ A \Leftrightarrow B$ which can be rewritten as $ (A \land B) \lor (\lnot A \land \lnot B)$
Antivalence: $ A \not\equiv B$ which can be rewritten as $ (A \land \lnot B) \lor (\lnot A \land B)$
Implication: $ A \Rightarrow B$ which can be rewritten as $ \lnot A \lor B$

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
\includegraphics[width=0.25\moimagesize]{din_und} \includegraphics[width=0.25\moimagesize]{din_oder} \includegraphics[width=0.25\moimagesize]{din_exor}
Negation Implication Equivalence
\includegraphics[width=0.25\moimagesize]{din_not} \includegraphics[width=0.25\moimagesize]{din_implikation} \includegraphics[width=0.25\moimagesize]{din_gleich}
(Authors: Hörner/Abele)

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  automatically generated 10/31/2008