The book I'm reading (Nadir Jeevanjee (auth.)-An Introduction to Tensors and Group Theory for Physicists-Birkhäuser Basel (2015)) defined an antisymmetric tensor of type $(r,0)$ or $(0,r)$ as
"one whose value changes sign under transposition of any two of its arguments."
OK, I totally understand the definition, but now the book is telling me that all tensors of type $(1,0)$, and $(0,1)$, are antisymmetric (the Wikipedia article about antisymmetric tensors agrees and even says they are trivial examples of an antisymmetric tensor). But I just don't get it! First, in the definition the book says "under transposition of any two of its arguments", but if the tensor is of type $(1,0)$, how can I transpose two arguments when I can just plug one? The only way, in my conception, for a tensor of type $(1,0)$ or $(0,1)$ to be antisymmetric is only if it is the $0$ tensor, since $0 = -0$.
What am I getting wrong?