3.NounA binary function from [0,1] × [0,1] to [0,1] which is defined in terms of the t-norm as follows:
x
→
y
=
sup
{
z
|
z
∗
x
⩽
y
}
{\displaystyle x\rightarrow y=\sup\{z|z*x\leqslant y\}}
, where
∗
{\displaystyle *}
denotes the t-norm function and
sup
{\displaystyle \sup }
denotes the supremum."The residuum of the minimum t-norm is a function
→
{\displaystyle \rightarrow }
such that y\\1,&{\text{if }}x\leqslant y.\end{cases}}}">
x
→
y
=
{
y
,
if
x
>
y
1
,
if
x
⩽
y
.
{\displaystyle x\rightarrow y={\begin{cases}y,&{\text{if }}x>y\\1,&{\text{if }}x\leqslant y.\end{cases}}}
y\\1,&{\text{if }}x\leqslant y.\end{cases}}}" />"