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Definition: A value x is a lower bound of S, if it is less than or equal to any element of S:
x is lower bound of S w.r.t. <= : <=> forall y in S: x <= y.
A value x is an upper bound of S, if it is greater than or equal to any element of S:
x is upper bound of S w.r.t. <= : <=> forall y in S: y <= x.
A lower (upper) bound of a set S need not be element of S.