3

I am having difficulty understanding (also because of my native language) the expression that is widely used in mathematics:

"Uniquely Determined".

For example:

$1)$ Cat theorem asserts that the color of the cat $C$ is uniquely determined by its breed $b(C)$.

In this example $ 1) $ it is correct to understand that each breed of cat determines a single color? That is, cats of the race $ X $ can only be, say, blue and never of another color?

And the reciprocal, is it also true? That is, if we have a green cat, can we say that it can only be (solely) a single specific breed?

An example, minus "Alice in Wonderland", the following example is really what interests me:

$2)$"(...) The previously given forms of Torelli's theorem follow at onlce from the remark that the symmetric product $(X)^{(g-1)}$ is birationally equivalent to the canonical $\Theta$-divisor on $J(X)$, and the fact, proved by A. Weil, that the canonical polarization determines uniquely the $\Theta$-divisor."

In this example $2)$ it is correct to state that: the canonical polarization of $J(X)$ determines a single $\Theta$-divisor? Is it also reciprocal? That is, given a $\Theta$-divisor to it is associated a single polarization?

Thank You!

Manoel
  • 2,216
  • 2
    Just because all cats of each breed have the same color doesn't mean that there can't be two breeds that share a color. – hmakholm left over Monica Aug 28 '18 at 20:04
  • 2
    A more elementary mathematical example: If $x,y\in\mathbb R$, then the sign of $x\cdot y$ is uniquely determined by the combination of the sign of $x$ the sign of $y$. But, being told that a particular product has sign $-$ does not tell you whether the signs of $x$ and $y$ are $-,+$ or $+,-$. – hmakholm left over Monica Aug 28 '18 at 20:06

1 Answers1

5

To say that $A$ is uniquely determined by $B$ is just to say that $A$ is a function of $B$. Note that if $A$ is uniquely determined by $B$, the converse need not follow. (It follows only when that function is invertible.)

For example, the sum of the squares of the eigenvalues of a $2\times 2$ matrix is uniquely determined by the determinant and trace, but the converse is not true.

symplectomorphic
  • 18,604
  • Okay. I understand what you say. But in the two examples that I put in my question, I ask two questions (affirmations). One I have already understood is not necessarily true. But the other statement I make is true ??? – Manoel Aug 28 '18 at 23:49