An if equivalence
is shown in logical statement, it means that it both the statements are logically same but we refer to it as XOR
gate(which is 1 when the inputs are different). Shouldn't it be XNOR
gate representing equivalence?
Why is equivalence in logic said to work similar as a XOR gate?
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0The English here isn't exactly clear. Also, what logically equivalence should get denoted as depends on which operations you take as primitives, and which notation you work in. If you take "XOR" as a primitive, then logical equivalence makes sense as "XNOR". But, usually logical equivalence gets taken as a primitive, so "XNOR" seems strange in that context. This isn't, in my opinion, so much a matter of math or logic though. – 2011-09-04
3 Answers
That's probably why it was said that it is similar. The pattern is the same, but the ones and zeros are reversed.
Yes, XNOR would be more appropriate. If you tell us which "we" refers to it as an XOR gate, there might be more to say.
Well, an XOR gate with inputs $A$ and $B$ can be described by the table:
$ \begin{array}{ c | c || c | } A & B & A \textrm{ XOR } B \\ \hline 0 & 0 & 0 \\ 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{array} $
whilst $A \Leftrightarrow B$ has truth table: $ \begin{array}{ c | c || c | } A & B & A \Leftrightarrow B \\ \hline F & F & T \\ F & T & F \\ T & F & F \\ T & T & T \end{array} $
If you interpret 0 as "false" and 1 as "true", then, yes, an inverted XOR gate would be more appropriate. But if you interpret 0 as "true" and 1 as "false", then a regular XOR gate will do.
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4On the other hand, the the names "AND", "OR" and so forth only make sense $u$nder the convention that 0 is false and 1 is tr$u$e. – 2011-09-04