In my linear algebra textbook, field is defined like below:
A field F is a set on which two operations $+$ and $\cdot$ (addition and multiplication, respectively) are defined so that, for each pair of elements $x,y$ in $F$, there are unique elements $x+y$ and $x\cdot y$ in $F$ for which following conditions hold for all elements $a,b,c$ in F
(the rest omitted)
What does it mean unique elements in here? It means $x+y$ and $x\cdot y$ is distinct?