Proving Monotonicity of t-norm

Question

For a lecture task I am trying to prove the monotonicity of a t-norm;

$$ T_H(x,y)=\frac{x\cdot y}{x +y -xy} $$

So I interpret this as being required to demonstrate that;

$$ T(x,y) \leq T(x,z) \textrm{ if } y \leq z $$

I am a mechanical engineer and not experienced with proving such things. It is a question for a lecture so I don't expect a full answer, but could anyone prod me in the right direction, using as simple an approach as possible? I am just struggling to get started with it. Thanks.

Answer 1

Rewrite the t-norm as $$T_H(x,y)=\frac{1}{x^{-1}+y^{-1}-1}.$$

So it increases monotonically with both $x$ and $y\in(0,1]$. When dealing with t-norms, try to put them into a form in which the associativity is kind of obvious. For example, $$T_H(x,y,z)=T_H(x,T_H(y,z))=T_H(T_H(x,y),z)=\frac{1}{x^{-1}+y^{-1}+z^{-1}-2}.$$

You see the pattern.