Elegant Notation: Intersection of Inequalities

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In summary, the conversation discusses the notation used to denote the solution set of inequalities and whether there is a simpler or more elegant way to express it. It is suggested that the notation \{inequality\} may not be the most accurate way to represent the solution set. Instead, the notation \{x | inequality\} is proposed as a more appropriate way to denote the set of x's that satisfy the given inequality. The conversation concludes with a revised notation for the union of solution sets.
  • #1
bomba923
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Is the notation
[tex]\left\{ \begin{gathered}
{\text{inequality}} \hfill \\
{\text{inequality}} \hfill \\
{\text{inequality}} \hfill \\
\end{gathered} \right\} [/tex]
generally accepted to denote the solution set (i.e., intersection) of the inequalities?

If so, then the following (part of a problem I came up with) should be easy to understand:
[tex]\bigcup\limits_{\begin{subarray}{l}
\left( {i,j,k} \right) \in \mathbb{N}^3 , \\
i < j < k \leqslant n
\end{subarray}} {\left\{ \begin{gathered}
\left( {y - y_i } \right)\left( {x_i - x_j } \right)\left| {\begin{array}{*{20}c}
{x_i - x_k } & {y_i - y_k } \\
{x_i - x_j } & {y_i - y_j } \\

\end{array} } \right| \geqslant \left( {x - x_i } \right)\left( {y_i - y_j } \right)\left| {\begin{array}{*{20}c}
{x_i - x_k } & {y_i - y_k } \\
{x_i - x_j } & {y_i - y_j } \\

\end{array} } \right| \hfill \\
\left( {y - y_i } \right)\left( {x_i - x_k } \right)\left| {\begin{array}{*{20}c}
{x_i - x_j } & {y_i - y_j } \\
{x_i - x_k } & {y_i - y_k } \\

\end{array} } \right| \geqslant \left( {x - x_i } \right)\left( {y_i - y_k } \right)\left| {\begin{array}{*{20}c}
{x_i - x_j } & {y_i - y_j } \\
{x_i - x_k } & {y_i - y_k } \\

\end{array} } \right| \hfill \\
\left( {y - y_j } \right)\left( {x_j - x_k } \right)\left| {\begin{array}{*{20}c}
{x_j - x_i } & {y_j - y_i } \\
{x_j - x_k } & {y_j - y_k } \\

\end{array} } \right| \geqslant \left( {x - x_j } \right)\left( {y_j - y_k } \right)\left| {\begin{array}{*{20}c}
{x_j - x_i } & {y_j - y_i } \\
{x_j - x_k } & {y_j - y_k } \\

\end{array} } \right| \hfill \\
\end{gathered} \right\}} [/tex]

Is there a simpler/more elegant way to express this, or is it fine the way it is?
 
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  • #2
That looks like a set of inequalities rather than the solution set to me. Rather like saying that {x2- 4= 0} denotes the set {2, -2}. (The second way is more "elegant"!)
 
  • #3
HallsofIvy said:
That looks like a set of inequalities rather than the solution set to me. Rather like saying that {x2- 4= 0} denotes the set {2, -2}. (The second way is more "elegant"!)
Well, to be more clear, perhaps I should add intersection symbols:
[tex]
\bigcup\limits_{\begin{subarray}{l}
\left( {i,j,k} \right) \in \mathbb{N}^3 , \\
i < j < k \leqslant n
\end{subarray}} {\left( \begin{gathered}
\left( {y - y_i } \right)\left( {x_i - x_j } \right)\left| {\begin{array}{*{20}c}
{x_i - x_k } & {y_i - y_k } \\
{x_i - x_j } & {y_i - y_j } \\

\end{array} } \right| \geqslant \left( {x - x_i } \right)\left( {y_i - y_j } \right)\left| {\begin{array}{*{20}c}
{x_i - x_k } & {y_i - y_k } \\
{x_i - x_j } & {y_i - y_j } \\

\end{array} } \right| \cap \hfill \\
\left( {y - y_i } \right)\left( {x_i - x_k } \right)\left| {\begin{array}{*{20}c}
{x_i - x_j } & {y_i - y_j } \\
{x_i - x_k } & {y_i - y_k } \\

\end{array} } \right| \geqslant \left( {x - x_i } \right)\left( {y_i - y_k } \right)\left| {\begin{array}{*{20}c}
{x_i - x_j } & {y_i - y_j } \\
{x_i - x_k } & {y_i - y_k } \\

\end{array} } \right| \cap \hfill \\
\left( {y - y_j } \right)\left( {x_j - x_k } \right)\left| {\begin{array}{*{20}c}
{x_j - x_i } & {y_j - y_i } \\
{x_j - x_k } & {y_j - y_k } \\

\end{array} } \right| \geqslant \left( {x - x_j } \right)\left( {y_j - y_k } \right)\left| {\begin{array}{*{20}c}
{x_j - x_i } & {y_j - y_i } \\
{x_j - x_k } & {y_j - y_k } \\

\end{array} } \right| \hfill \\
\end{gathered} \right)} [/tex]
Essentially, I wish to denote a union of solution sets :shy:
Would that be clear/understood from the way I rewrote it here?
 
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  • #4
It looks like you are still thinking of an inequality as the set of things that satisfy it, this isn't the usual use of the notation. The set of x's that satisfy x>2 would be written:

{x| x>2}

(specifiy some set x is coming from if it's not clear from the context, like x a real number, or integer), not

x>2
 
  • #5
Thanks shmoe :smile:
In that case, this would be the more correct way to denote my union of solution sets:
[tex]\bigcup\limits_{\begin{subarray}{l}
\left( {i,j,k} \right) \in \mathbb{N}^3 , \\
i < j < k \leqslant n
\end{subarray}} {\left\{ {\left( {x,y} \right)\left| \begin{gathered}
\left( {y - y_i } \right)\left( {x_i - x_j } \right)\left| {\begin{array}{*{20}c}
{x_i - x_k } & {y_i - y_k } \\
{x_i - x_j } & {y_i - y_j } \\

\end{array} } \right| \geqslant \left( {x - x_i } \right)\left( {y_i - y_j } \right)\left| {\begin{array}{*{20}c}
{x_i - x_k } & {y_i - y_k } \\
{x_i - x_j } & {y_i - y_j } \\

\end{array} } \right| \hfill \\
\left( {y - y_i } \right)\left( {x_i - x_k } \right)\left| {\begin{array}{*{20}c}
{x_i - x_j } & {y_i - y_j } \\
{x_i - x_k } & {y_i - y_k } \\

\end{array} } \right| \geqslant \left( {x - x_i } \right)\left( {y_i - y_k } \right)\left| {\begin{array}{*{20}c}
{x_i - x_j } & {y_i - y_j } \\
{x_i - x_k } & {y_i - y_k } \\

\end{array} } \right| \hfill \\
\left( {y - y_j } \right)\left( {x_j - x_k } \right)\left| {\begin{array}{*{20}c}
{x_j - x_i } & {y_j - y_i } \\
{x_j - x_k } & {y_j - y_k } \\

\end{array} } \right| \geqslant \left( {x - x_j } \right)\left( {y_j - y_k } \right)\left| {\begin{array}{*{20}c}
{x_j - x_i } & {y_j - y_i } \\
{x_j - x_k } & {y_j - y_k } \\

\end{array} } \right| \hfill \\
\end{gathered} \right.} \right\}} [/tex]
Right?
 
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  • #6
Sure, I'd call that more correct, but you'd probably want to put some "or"'s in there.
edit-spelling, I wasn't calling you "Sue"
 
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  • #7
shmoe said:
Sue, I'd call that more correct, but you'd probably want to put some "or"'s in there.
Some "or's" ? Why so?
 
  • #8
maybe "and"'s then. You just have a list of inequalities as conditions,

{x|x>2, x>4, x=65}

What does this mean? It's ambiguous as it's written.
 
  • #9
shmoe said:
maybe "and"'s then. You just have a list of inequalities as conditions,

{x|x>2, x>4, x=65}

What does this mean? It's ambiguous as it's written.
Do you mean like this:
[tex]\bigcup\limits_{\begin{subarray}{l}
\left( {i,j,k} \right) \in \mathbb{N}^3 , \\
i < j < k \leqslant n
\end{subarray}} {\left\{ {\left( {x,y} \right)\left| \begin{gathered}
\left( {y - y_i } \right)\left( {x_i - x_j } \right)\left| {\begin{array}{*{20}c}
{x_i - x_k } & {y_i - y_k } \\
{x_i - x_j } & {y_i - y_j } \\

\end{array} } \right| \geqslant \left( {x - x_i } \right)\left( {y_i - y_j } \right)\left| {\begin{array}{*{20}c}
{x_i - x_k } & {y_i - y_k } \\
{x_i - x_j } & {y_i - y_j } \\

\end{array} } \right| \wedge \hfill \\
\left( {y - y_i } \right)\left( {x_i - x_k } \right)\left| {\begin{array}{*{20}c}
{x_i - x_j } & {y_i - y_j } \\
{x_i - x_k } & {y_i - y_k } \\

\end{array} } \right| \geqslant \left( {x - x_i } \right)\left( {y_i - y_k } \right)\left| {\begin{array}{*{20}c}
{x_i - x_j } & {y_i - y_j } \\
{x_i - x_k } & {y_i - y_k } \\

\end{array} } \right| \wedge \hfill \\
\left( {y - y_j } \right)\left( {x_j - x_k } \right)\left| {\begin{array}{*{20}c}
{x_j - x_i } & {y_j - y_i } \\
{x_j - x_k } & {y_j - y_k } \\

\end{array} } \right| \geqslant \left( {x - x_j } \right)\left( {y_j - y_k } \right)\left| {\begin{array}{*{20}c}
{x_j - x_i } & {y_j - y_i } \\
{x_j - x_k } & {y_j - y_k } \\

\end{array} } \right| \hfill \\
\end{gathered} \right.} \right\}} [/tex]
Correct?
 
  • #10
Yes, post #9 is correct.
 

FAQ: Elegant Notation: Intersection of Inequalities

What is elegant notation?

Elegant notation is a mathematical concept that refers to a concise and aesthetically pleasing way of writing mathematical equations or statements. It is often used to simplify complex ideas and make them easier to understand.

What is the intersection of inequalities?

The intersection of inequalities is the solution set that satisfies two or more inequality equations simultaneously. It is represented by the overlapping region on a graph or the common values between the equations.

How is the intersection of inequalities represented in elegant notation?

The intersection of inequalities is commonly represented using set notation, where the solution set is enclosed in curly brackets. For example, if the inequalities are x ≤ 5 and x ≥ 2, the intersection would be represented as {x | 2 ≤ x ≤ 5}.

What is the purpose of using elegant notation in mathematics?

The purpose of elegant notation is to simplify complex mathematical ideas and make them easier to understand. It also allows for more efficient and concise communication among mathematicians and scientists.

Can elegant notation be applied to other areas of science?

Yes, elegant notation can be applied to various areas of science, such as physics, biology, and chemistry. It is a universal language that allows scientists to communicate complex ideas in a concise and clear manner.

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