# Can two free variables in lambda calculus have different values?

I am studying lambda calculus for the first time and I was trying to do the reduction beta of the lambda term $$(\lambda x.xy)y$$. Can I assume that these two free variables $$y$$ are the same? Or do I need to rename the rightmost $$y$$?

$$y$$ is free in the term you have given, so it is indeed the same variable.
You would not have the same variable in case $$y$$ was bound by the lambda. That would be the case, for example, if the term were to be $$(\lambda y.xy)y$$; then the rightmost $$y$$ would be free, but the one in $$xy$$ would be bound by lambda and, hence, a different variable.