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A mathematical expression is feasible, if it obeys the following rules:

  • Any real number may be used
  • One may use brackets "(" and ")" to structure the expression, and to make it well-defined
  • The allowed mathematical operations are addition ($+$), subtraction ($-$), multiplication ($*$).
  • Furthermore there is the special operation plus-minus ($\pm$).

Every feasible expression encodes the following set of numbers: every occurrence of $\pm$ in the expression is replaced once by $+$ and once by $-$, in all possible combinations.

Examples: For the expression $(1\pm2)\pm1$, we get the four corresponding numbers $(1+2)+1=4$ and $(1+2)-1=2$ and $(1-2)+1=0$ and $(1-2)-1=-2$; hence this expression encodes the set $\{-2,0,2,4\}$. Similarly one sees that the expression $(4\pm\frac12)\pm\frac12$ encodes the three-element set $\{3,4,5\}$.

Question: Does there exist a feasible expression that encodes the set $\{1,2,4\}$?

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up vote 10 down vote

Another, slightly more symmetrical option:

$(1.5 \pm 0.5) * (1.5 \pm 0.5)$

Explanation:

$(1.5 \pm 0.5)$ is equal to 1 or 2. So we have 1*1, 1*2, 2*1, and 2*2.

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up vote 7 down vote

A set containing any single value is obviously a feasible expression. Further, given any set S of numbers which can be encoded as a feasible expression E, and any number x, the expression $x+(\frac12±\frac12)((E)-x)$ will be a feasible expression containing all of the items in S, along with x, but nothing else. Thus, any set of real values may be encoded as a feasible expression. Because {1,2,4} is a set of real values, it may be encoded as a feasible expression using the above approach, e.g.

Start with first item: $1$

Add second: $2 + (\frac12±\frac12)(1-2)$, i.e. $1\frac12±\frac12$

Add third: $4+(\frac12±\frac12)((1\frac12±\frac12)-4)$, i.e. $4+(\frac12±\frac12)(-2\frac12±\frac12)$

That approach will not necessarily yield the most compact expression for any given set of numbers, but it will find an expression meeting the criteria.

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That is quite beautiful. – Scott M 12 hours ago
    
“[…] any set of real values may be encoded as a feasible expression” There is a “finite” missing somewhere in there. I’m pretty sure you can’t find an expression for R \ Q or even N. – Édouard 12 hours ago
    
@Édouard If feasible expressions must be finite (not specified), then you are correct. However, if infinitely long expressions are allowed, then I believe that one could use this method to generate any countable set of real numbers. In all cases though, I believe you are right as far as uncountable sets go. – Scott M 12 hours ago
up vote 5 down vote

We can use:

$1 + \underbrace{(\frac12\pm\frac12)}_{\text{$0$ or $1$}}*\underbrace{(2\pm1)}_{\text{$1$ or $3$}}$
Which gives us the corresponding numbers $1 + 0*1$, $1 + 0*3$, $1 + 1*1$ and $1 + 1*3$
And hence it encodes the three-element set $\{1, 2, 4\}$

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