Express Sentences Using Mathematical symbols

Question

I know this supposed to be relatively simple but I'm not sure if I am doing the questions right. I am relatively new to logic and abstract math. I have attached the questions and my attempts at a solution. Any help would be much welcomed.

Suppose that $A\subset \mathbb{R}$ is an interval and that $f:A\to \mathbb{R}$ is a function. Write the following two sentences using mathematical symbols:

1. The function is not a constant function.
2. The function does not take the same value twice.

My attempt:

1. $\forall B\in\mathbb{R},\exists x\in A, f(x)\ne B$.
2. $\forall x\in A, f(x_1)=f(x_2) \rightarrow x_1 = x_2$.

Answer 1

As noted in the comments, (1) looks fine, though I really like HenningMakholm's slightly modified

$\neg(\exists B\in\mathbb{R})(\forall x\in A)f(x)=B$.

For (2), the conclusion is correct, i.e. $f(x_1) = f(x_2) \to x_1 = x_2$. However, you did not ever tell us where $x_1$ and $x_2$ live. Since we want this statement to hold true for all $x_1$ and $x_2$, let's say so! Hence I might propose the solution

$(\forall x_1,x_2\in A) (f(x_1)=f(x_2)\to x_1=x_2)$.

Alternatively, we could use the idea that if we fix $x_1$ first, then go searching for an $x_2$, we are never going to find one that gives the same value when evaluating $f$. Perhaps something like

$(\forall x_1\in A)\neg(\exists x_2\in A)(x_1 \ne x_2 \land f(x_1)=f(x_2))$.

If I have correctly formatted things (these notation-heavy, "English free" sentences always give me a headache), we should be able to read this as "For every $x_1\in A$, there does not exist an $x_2\in A$ such that both $x_1\ne x_2$ and $f(x_1) = f(x_2)$." That is, given any $x_1$, we cannot produce an $x_2$ different from $x_1$ that gives a different value of $f$. If we are happy working with set operations, we might also write

$(\forall x_1\in A)\neg(\exists x_2\in A\setminus\{x_1\})f(x_1)=f(x_2)$.