Difference between sup and max

Question

I am working at some Fuzzy-Logic and I am having my problems with the inferece. While using the generalised modus ponens you are using this formula

μB'(y) := sup{min(μA'(x),min(μA(x),μB(y))) | x∈X} for y∈Y

My Question is, where is the Difference between the min/max Operators and the sup/inf Operators? Aren't they both just for finding the largest/smallest Value?

I tried, but the tags I tied did not exist, things like inference, Minimum-Maximum — yoko, Nov 30 '15 at 10:55
See Infimum and supremum : min and max of a set X (if they exist) are "into" $X$ while sup-inf are not necessarily so. — Mauro ALLEGRANZA, Nov 30 '15 at 10:58
See in Wiki the "classical" example; $0$ is the inf for the set $\mathbb R^+$ of positive real numbers, but $0$ it is not positive, and thus $0 \notin \mathbb R^+$. — Mauro ALLEGRANZA, Nov 30 '15 at 11:01

score 1 · Accepted Answer · answered Nov 30 '15 at 11:00

They only differ when the set on which they operate has an infinite number of elements:

If X has a finite number of elements, then $$\sup (x\ :\ x\in X)=\max (x\ :\ x\in X)$$ Because the sup is then simply attained at some point in X. However, this is not necessarily the case, as the following example will illustrate: Let $X=(0,1) $. Then $\sup (X)=1$, while 1 is not a member of X. $\max (X) $ does not exists in this case. Hope this helps :)

Difference between sup and max

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