Need help with writing set notation.

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I need help with some set notation. I am an engineering major and don't know the proper notation. I am trying to write two statements. I have two sets, $x_1$ = {1,2,3}, and $x_2$ = {4,5,6}.

Verbal Statements:
$y_1$ was measured for all values of $x_1$ and $x_2$
$y_2$ was measured, for all values of $x_1$, and only the first element (4) of $x_2$

Attempt:
$y_1$ measured $\forall x_1$, $\forall x_2$
$y_2$ measured $\forall x_1$, {4} $\cap\: x_2$
or
$y_2$ measured $\forall x_1$, {4} $\in x_2$

Are any of these correct? Is there a more proper way to write this?

 

[Stephen Tashi]

I am trying to write two statements.

You should explain in what context you are trying to write something. Is it in a technical report? Homework for a logic course? Homework for a statistics course? After all, you can express yourself using only words. In many contexts, mathematical notation is unnecessary.

 

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Stephen,

Thanks for the reply. It is for a technical paper (journal publication). I prefer to use the set notation because it is going in a table with limited space.

 

[Stephen Tashi]

The safest procedure is to look at tables from articles the journal published and see what those authors did.

To me, it is confusing to have integers both as subscripts on the sets $x_i$ and also as to have them as elements of the sets themselves. If an integer such as 4 is only going to appear as an element of a single set $x_i$, why don't you simply list the elements where the measurements were taken? Such as {1,2,3,4} ?

 

[blink-]

Hey Stephen,

I was trying to use a simplified example. The sets contain floats and integers. Here is a picture of the table in question.



I would prefer using set notation, but if you think listing all elements is a better idea, I can do that.

 

[Stephen Tashi]

I think this is qestion of "literary style", not a question of how to write mathematics. To me, it is better to write "all values of $x_1$" instead of using the mathematical abreviation "for all" and not usiing other consistent notation with it. To use a quantifier, you need a variable. Your idea would be expressed as "for all v such that v is a member of $x_1$". The quantifer would be applied to a variable, not to a set.

if you want to use $x_1, x_2$ as variables instead of sets, you could use notation like
$x1 \in \{-90.0 , -45.0, 0, 45.0 , 90 \} , x_2 \in \{12.6\}$

 

[blink-]

Thanks for all the replies Stephen. You have been a big help. I do agree, this is more of a literary style question, sorry if this was the wrong forum to post it in.

Just to confirm, you think something like this would be best:

x₁ angular position (deg)
x₂ gas velocity (cm/s)

y₁ frequency (Hz), observed at x₁ $\in$ {-90,-45,0,45,90}, x₂ $\in$ {5.5,7,11,12.6}
y₂ fraction (-), observed at x₁ $\in$ {-90,-45,0,45,90}, x₂ $\in$ {12.6}

 

[Stephen Tashi]

I can't think of a better way than what you wrote.

Again, papers published in a journal are the best guide to what the journal's editors like.