Express Sentences Using Mathematical symbols 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:
  
  
*
  
*The function is not a constant function.
  
*The function does not take the same value twice.
  

My attempt:


*

*$\forall B\in\mathbb{R},\exists x\in A, f(x)\ne B$.

*$\forall x\in A, f(x_1)=f(x_2) \rightarrow x_1 = x_2$.

 A: 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

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

