How would a statement beginning, "For all $x$ in set $S$ except $x=a$..." be translated into symbolic predicate logic? I'm somewhat of a purist in symbols and am less than satisfied with "$\forall x \in S$ except $x = a$..." or "$\forall x \in S$ where $x \ne a$...".
Symbolic predicate logic for "for all elements in a set except this one..."
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$\begingroup$
logic
elementary-set-theory
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1As the comment on one of the answers suggests, the question in the title is not the same as the question in the body. The title says "except one...", which may be any one, and need not be *a* in particular: it might be *b* or *c* or .... It is more complicated to express "all elements except exactly one" than it is to express "all except this particular one." – 2012-01-12
3 Answers
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I would use $(\exists a\in S)(\forall x\in S)(x\neq a \Leftrightarrow \cdots)$. It's not a direct correspondence to the English, but not everything in symbolic logic is.
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0More precisely, it is$a$valid interpretation of the title the question had when this answer was written (and still had when I wrote my comment). Then it said: "for all elements in a set except one..." – 2012-01-15
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$\forall x \in S\setminus \{a\}, \,\,\,\ldots$
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The exact correspondence in first order language is as follows:
$∀x((x∈S ∧ x≠a) → ⋯)$
It reads: for all $x$, if $x$ belongs to $S$ and $x$ is not $a$, then....
$x≠a$ is just a short form for $¬(x=a)$
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0Certainly one can embed any desired information within the $\dots$. My comment had to do with the fact that the informal meaning of "all except $a$ have property $P$" probably includes that $a$ does not have property $P$. – 2012-01-12