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| A bracketed list of propositional expressions in the form <math>\texttt{(} e_1 \texttt{,} e_2 \texttt{,} \ldots \texttt{,} e_{k-1} \texttt{,} e_k \texttt{)}</math> indicates that exactly one of the propositions <math>e_1, e_2, \ldots, e_{k-1}, e_k</math> is false.

| A parenthesized sequence of propositional expressions in the form <math>\texttt{(} e_1 \texttt{,} e_2 \texttt{,} \ldots \texttt{,} e_{k-1} \texttt{,} e_k \texttt{)}</math> indicates that exactly one of the propositions <math>e_1, e_2, \ldots, e_{k-1}, e_k</math> is false.

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| A concatenation of propositional expressions in the form <math>e_1\ e_2\ \ldots\ e_{k-1}\ e_k</math> indicates that all of the propositions <math>e_1, e_2, \ldots, e_{k-1}, e_k</math> are true, in other words, that their [[logical conjunction]] is true.

| A concatenated sequence of propositional expressions in the form <math>e_1\ e_2\ \ldots\ e_{k-1}\ e_k</math> indicates that all of the propositions <math>e_1, e_2, \ldots, e_{k-1}, e_k</math> are true, in other words, that their [[logical conjunction]] is true.

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All other propositional connectives can be obtained in a very efficient style of representation through combinations of these two forms.  Strictly speaking, the concatenation form is dispensable in light of the bracketed form, but it is convenient to maintain it as an abbreviation of more complicated bracket expressions.  While working with expressions solely in propositional calculus, it is easiest to use plain parentheses for logical connectives.  In contexts where ordinary parentheses are needed for other purposes an alternate typeface <math>\texttt{(} \ldots \texttt{)}</math> may be used for logical operators.

All other propositional connectives can be obtained through combinations of these two forms.  Strictly speaking, the parenthesized form is sufficient to define the concatenated form, making the latter formally dispensable, but it is convenient to maintain it as a concise way of expressing more complicated combinations of parenthesized forms.  While working with expressions solely in propositional calculus, it is easiest to use plain parentheses for logical connectives.  In contexts where ordinary parentheses are needed for other purposes an alternate typeface <math>\texttt{(} \ldots \texttt{)}</math> may be used for logical operators.

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The briefest expression for logical truth is the empty word, usually denoted by <math>\varepsilon\!</math> or <math>\lambda\!</math> in formal languages, where it forms the identity element for concatenation.  To make it visible in this text, it may be denoted by the equivalent expression <math>{}^{\backprime\backprime} \texttt{((~))} {}^{\prime\prime},</math> or, especially if operating in an algebraic context, by a simple <math>{}^{\backprime\backprime} 1 {}^{\prime\prime}.</math>  Also when working in an algebraic mode, the plus sign <math>{}^{\backprime\backprime} + {}^{\prime\prime}</math> may be used for [[exclusive disjunction]].  For example, we have the following paraphrases of algebraic expressions by bracket expressions:

The simplest expression for logical truth is the empty word, usually denoted by <math>\varepsilon\!</math> or <math>\lambda\!</math> in formal languages, where it forms the identity element for concatenation.  To make it visible in context, it may be denoted by the equivalent expression <math>{}^{\backprime\backprime} \texttt{((~))} {}^{\prime\prime},</math> or, especially if operating in an algebraic context, by a simple <math>{}^{\backprime\backprime} 1 {}^{\prime\prime}.</math>  Also when working in an algebraic mode, the plus sign <math>{}^{\backprime\backprime} + {}^{\prime\prime}</math> may be used for [[exclusive disjunction]].  For example, we have the following paraphrases of algebraic expressions by means of parenthesized expressions:

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It is important to note that the last expressions are not equivalent to the triple bracket <math>\texttt{(} a \texttt{,} b \texttt{,} c \texttt{)}.</math>

It is important to note that the last expressions are not equivalent to the 3-place parenthesis <math>\texttt{(} a \texttt{,} b \texttt{,} c \texttt{)}.</math>

===Differential Expansions of Propositions===

===Differential Expansions of Propositions===