In my book, it says the above set fails the first axiom. It says if I take two sets $p_1(x)=x^3-x^2$ and $q(x)= 1-x^2$. They are not closed under addition. I can understand why that's true by using a specific example. But for me, if I get a question like this on a test, I won't be able to think of a specific example.
My question is how to prove axiom 1 fails by a general approach. I need an approach that I could use even if the question asks to prove some other set as a vector space. My approach
$$p_1(x)=ax^3+bx^2+cx+d p_2(y)=ay^3+by^2+cy+d$$
$$p_1(x)+p_2(y)= a(x^3+y^3)+ b(x^2+y^2)+c(x+y)+2d$$
If I use a general example, aren't the polynomials closed under addition because my sum is also a third-degree polynomial?
Best Answer
The Axiom you are trying to disprove says:
So you want to establish its negation. The negation says:
The way to prove this is to establish the existence of such $p$ and $q$. The best way of doing so is to exhibit a $p$ and a $q$ that satisfy this statement; that is, to give an example. So the “general method” to prove this is precisely to give an example.
So a better way to phrase what you are asking (or a better question to ask) is: “How could I try to come up with an example if I don’t already know one?”
Well, first, your computations are off because your $p$ and $q$ are in different sets: one is a polynomial in $x$, the other is a polynomial in $y$. And worse, for some reason you are assuming that they have the same coefficients! Those are not two arbitrary polynomials of degree $3$ in the same set.
So, take two arbitrary polynomials of degree $3$ in $x$. Say, $p$ and $q$. Say: $$\begin{align*} p(x) &= ax^3 + bx^2 + cx + d\\ q(x) &= \alpha x^3 + \beta x^2 + \gamma x + \delta, \end{align*}$$ Where $a,b,c,d,\alpha,\beta,\gamma,\delta$ are some specific (but right now unspecified) real numbers. Note that in order to be polynomials of degree $3$, we need $a\neq 0$ and $\alpha\neq 0$.
Now, their sum is $$(p+q)(x) = (a+\alpha)x^3 + (b+\beta)x^2 + (c+\gamma)x + (d+\delta).$$ In order for this to be a polynomial of degree $3$, we need $a+\alpha\neq 0$.
So in order to find an example to exhibit the failure of Axiom 1, you want to pick some values of $a,b,c,d$ with $a\neq 0$, and some values of $\alpha,\beta,\gamma,\delta$, with $\alpha\neq 0$, but such that $a+\alpha=0$. At this point, presumably some specific values will suggest themselves, and you can go ahead and give them explicitly.
Advice: Do not stop at saying “but we cannot guarantee that $a+\alpha\neq 0$.” Actually go ahead and give an example showing that this can fail.
Note: In most cases for a set that is not a vector space, a lot of axioms are going to fail, not just one. So the fact that this one fails closure under addition, but also fails to have a zero vector, should not come as a surprise. If one thing goes wrong, often lots of things go wrong.
Final comment: Not everyone gives the axioms of a vector space the same way, or in the same order. In particular, closure under addition is not always listed as “axiom 1” (sometimes it is implicitly given by saying that vector addition “is an operation”). When asking questions about axioms of a vector space, be sure to give the Axiom you are talking about explicitly to make sure everyone is thinking about the same axiom.