So the teacher decided to make our life harder by giving us an extra-credit problem:
Use the language of the first-order predicate calculus to express that in a group $ S $ of elements with a binary operation ‘$ \cdot $’, there exists only one identity element.
Considering that he didn’t even cover anything like this in class, I’m breaking my brains trying to figure it out. Can someone explain this one to me?
The formula is $$ (\exists e)((\forall x)((x \cdot e = x) \land (e \cdot x = x)) \land (\forall f)((\forall x)((x \cdot f = x) \land (f \cdot x = x)) \Rightarrow (f = e))). $$ You want to say that there exists an identity element, i.e., an element $ e $ such that $ x \cdot e = x $ and $ e \cdot x = x $ for all $ x $, and if $ f $ is any other element satisfying the same two properties, then $ f = e $.