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Logical biconditional
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===Truth table=== The following is a truth table for <math>A \leftrightarrow B</math>: {{2-ary truth table|1|0|0|1|<math>A \leftrightarrow B</math>}} {{-}} When more than two statements are involved, combining them with <math>\leftrightarrow</math> might be ambiguous. For example, the statement :<math>x_1 \leftrightarrow x_2 \leftrightarrow x_3 \leftrightarrow \cdots \leftrightarrow x_n</math> may be interpreted as :<math>(((x_1 \leftrightarrow x_2) \leftrightarrow x_3) \leftrightarrow \cdots) \leftrightarrow x_n</math>, or may be interpreted as saying that all {{math|''x<sub>i</sub>''}} are ''jointly true or jointly false'': :<math>(x_1 \land \cdots \land x_n) \lor (\neg x_1 \land \cdots \land \neg x_n)</math> As it turns out, these two statements are only the same when zero or two arguments are involved. In fact, the following truth tables only show the same bit pattern in the line with no argument and in the lines with two arguments: [[File:Variadic logical XAND.svg|thumb|left|220px|<math>~x_1 \leftrightarrow \cdots \leftrightarrow x_n</math><br />meant as equivalent to<br /><math>\neg~(\neg x_1 \oplus \cdots \oplus \neg x_n)</math><br /><br />The central Venn diagram below,<br />and line ''(ABC )'' in this matrix<br />represent the same operation.]] [[File:Variadic logical all or nothing.svg|thumb|right|220px|<math>~x_1 \leftrightarrow \cdots \leftrightarrow x_n</math><br />meant as shorthand for<br /><math>(~x_1 \land \cdots \land x_n~)</math><br /><math>\lor~(\neg x_1 \land \cdots \land \neg x_n)</math><br /><br />The Venn diagram directly below,<br />and line ''(ABC )'' in this matrix<br />represent the same operation.]] {{-}} The left Venn diagram below, and the lines ''(AB )'' in these matrices represent the same operation.
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