Largest x for which an equation is true

[Cinitiator]

Homework Statement

How to use mathematical notation to express the largest x for which an equation is true?

For example, how to express the largest x for which A(x) = B(x)? I need to be able to give the instruction to find the largest x for which A(x) = B(x) is true, but I wonder if it can be written down in a more algebraic manner, instead of relying purely on language.

Homework Equations

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The Attempt at a Solution

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[Mark44]

Homework Statement

How to use mathematical notation to express the largest x for which an equation is true?

For example, how to express the largest x for which A(x) = B(x)? I need to be able to give the instruction to find the largest x for which A(x) = B(x) is true, but I wonder if it can be written down in a more algebraic manner, instead of relying purely on language.

Homework Equations

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The Attempt at a Solution

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max{x $\in$ R : A(x) = B(x)}

In words, this repesents the largest real number x for which A(x) = B(x). You didn't say, but I'm assuming you mean real number values.

Typically, this would be a set with a finite number of values in it, of which you want the largest. If the solution set of the equation A(x) = B(x) has infinitely members, there won't be a largest value.

 

[Cinitiator]

max{x $\in$ R : A(x) = B(x)}

In words, this repesents the largest real number x for which A(x) = B(x). You didn't say, but I'm assuming you mean real number values.

Typically, this would be a set with a finite number of values in it, of which you want the largest. If the solution set of the equation A(x) = B(x) has infinitely members, there won't be a largest value.

Thanks a lot for your input.

How is the type of notation which is used in this case called? What branch of mathematics does it belong to (especially the max operator)? Formal logic?

 

[HallsofIvy]

[itex]\{x\in R: A(x)= B(x)\}[itex] is "set buider notation". It specifices the set of all real numbers, x, such that A(x)= B(x). And the "max" in front is an operator that returns the largest member of the set.

 

[Cinitiator]

[itex]\{x\in R: A(x)= B(x)\}[itex] is "set buider notation". It specifices the set of all real numbers, x, such that A(x)= B(x). And the "max" in front is an operator that returns the largest member of the set.

Thanks for your input.

Is there any documentation of the max operator? Is it recognized in the mainstream mathematics?

 

[Mark44]

Thanks for your input.

Is there any documentation of the max operator? Is it recognized in the mainstream mathematics?

You are way overthinking this. When applied to a set for which it makes sense (finite set of elements that have an inherent ordering), the max of the set is the largest element.

This is well known in mathematics.

Putting on my moderator hat: You now have three or four threads, all asking more-or-less the same question. Do not start a new thread on this same subject.

 

[skiller]

Typically, this would be a set with a finite number of values in it, of which you want the largest. If the solution set of the equation A(x) = B(x) has infinitely members, there won't be a largest value.

Sorry for butting in, but did you actually mean to say that?

What if the solution set was:

$\{ x \in \mathbb{R} : 0 \leq x \leq 1 \}$

or

$\{ n \in \mathbb{Z} : n \leq 0 \}$

to take two very simple examples?

These two sets both have infinitely many members, but they each have a largest value (1 and 0, respectively).

Apologies if I have misunderstood.

 

[Mark44]

If the solution set of the equation A(x) = B(x) has infinitely members, there won't be a largest value.

Sorry for butting in, but did you actually mean to say that?

What if the solution set was:

$\{ x \in \mathbb{R} : 0 \leq x \leq 1 \}$

or

$\{ n \in \mathbb{Z} : n \leq 0 \}$

to take two very simple examples?

These two sets both have infinitely many members, but they each have a largest value (1 and 0, respectively).

Apologies if I have misunderstood.

Thanks for correcting me - you're absolutely right. I was thinking in terms of solutions to various kinds of equations, where the solutions are discrete, and didn't consider the possibility of a solution set that was bounded interval.

 

[skiller]

I was thinking in terms of solutions to various kinds of equations, where the solutions are discrete, and didn't consider the possibility of a solution set that was bounded interval.

Strange reply, but I'm sure you know what you're talking about!

 

[SteveL27]

Strange reply, but I'm sure you know what you're talking about!

The correct correction is: An infinite set of reals does not necessarily have a largest element. Of course some sets of reals do have a largest element.