According to Wikipedia, the axioms of additive valuation and multiplicative valuation requires $v(a+b)\geq \min\{v(a),v(b)\}$ and $v(a+b)\leq \max\{v(a),v(b)\}$, with equality if $v(a)\neq v(b)$.
What goes wrong if we are not necessarily allow the equality with $v(a)\neq v(b)$? (I mean $v(a+b)>\min\{v(a),v(b)\}$ and $v(a)\neq v(b)$)
Well, if $v$ is an additive valuation, $v(a)\neq v(b)$, and $v(a+b)>\min\{v(a),v(b)\}$, if we assume $v(a)=\min\{v(a),v(b)\}$,hence $v(\frac{a+b}{a})>0$, Where is the contradiction? And the same for multiplicative valuation.
Thanks in advance.