@user21820 I wanted to ask you something else. I have been studying linear optimization and there's this question "maximize $v.x$ subject to constraints $Ax+b \geq b$ where $A$ is a $m x n$ matrix, that is $m$ inequalities with $n$ variables. If this problem is reduced to a set of linear equations with $k$ variables, then what is the minimum value of $k$?"

@user21820 I was thinking that the answer is $n-m+1$ assuming we have more variables than equations. However, this answer is wrong. Can you help me with some hints and information to arrive at the right answer?

BTW I could not figure why the $$ sign is not converting text to Latex. I am sorry for the inconvenience.

@user21820 And I meant to type $Ax \geq b$ and not $Ax + b \geq b$. I am still getting used to typing math on chat. I just now figured out that we need to use ChatJax to type math in these comments.

I’m looking for an example of a relation whose arity is one. Can someone help me?

@ConGovDeIn A unary relation-symbol is simply a unary predicate-symbol. Not sure what you mean by "example"..

@TryingHardToBecomeAGoodPrSlvr There is no LaTeX in chat, and I don't use it either.

@TryingHardToBecomeAGoodPrSlvr I don't know what "reduce" means here. Ax≥b is clearly equivalent to a system of linear inequalities given by the definition of matrix multiplication.

@user21820 Ummm... like + is an example of binary function symbol, $^{-1}$ is an example of unary function symbol. What relation symbol we have whose arity is 1?

@ConGovDeIn Um, every time you use induction you are introducing a predicate symbol. Is that a good enough example?

Okay.

With your help last time, I have come up with a language for trigonometry: $$\mathcal{L} ~is~ \{a_0, a_1, a_2 \cdots , +, •, \sin , \cos, \tan, \leq\}$$ where $a_n$ refers to the nth natural number and $a_0$ stands for the number 0

But I’m thinking how would I take into account the rational and irrational numbers?

@user21820 I got the answer finally! It seems the question was basically meant to ask "if you wish to solve this optimization problem, how many variables do you need to solve for?" It assumes that there might be some redundant inequalities and we just select the ones that apply. The catch to was to know that you also need to solve the dual to know that we have solved the problem correctly.

It was poorly worded but I finally got the answer. Thanks for the nudge to help me think in the right direction!

@TryingHardToBecomeAGoodPrSlvr I see. Thanks for the update!

@ConGovDeIn So you're not going to have a constant-symbol for π?

Rational numbers can be obtained from integers, so you don't need constants for them.

Now, maybe your question is how to include enough irrational numbers that you might want to reason about when dealing with trigonometry.

As I said earlier:

> In your axiomatization you need at least the ordered field axioms for real numbers.

Maybe at this point it is too early for you to look at axioms, since you are only asking about the language.

But it is worth knowing that function-symbols and axioms allow you to get new objects from other objects without actually having constant-symbols for all the objects of interest.

If you choose to just have +,· for arithmetic, then the ordered field axioms allow you to construct all the rationals from just the constant-symbols 0,1. You don't even need a constant-symbol for each natural, unlike your attempt.

Trigonometry involves π a lot, so you probably want a constant-symbol for that since it's not rational.

Also, you may need things like sqrt. That kind of thing can be provided for by suitable additional axioms besides the ordered field axioms. Note that the ordered field axioms are satisfied by the rationals, so clearly we do need additional axioms if we want ∃x ( x^2=1+1 ) to be provable.

Again, it may be too early for you, but I'll just say that you probably want to have the axioms of real closed fields:

@ConGovDeIn: I actually think that now is not the right time for you to attempt any sort of formalization of basic trigonometry. I have no idea who or what suggested that to you, and I think you have gotten enough (even though vague) idea of the kind of function-symbols and constant-symbols you might want. I think that's good enough, because you can't do more without actually understanding FOL in full.