LMIs in Control/Matrix and LMI Properties and Tools/Set minimum theorem

In set theory and number theory, the set minimum theorem states that if S is a well-defined, constant finite set of positive integers, then for all sufficiently large positive integers x, the set S has at least one element that is smaller than x. Since S is a non-changing subset of positive integers, it has a minimum and that minimum is a constant; since it’s a finite constant, x simply has to exceed it. For example, if S = {12, 7, 9, 16, 13, 21}, a set with six terms, then min(S) = 9, so the statement “S has at least one term that is smaller than x” is false if x is between 1 and 8, inclusive, but it is true for x > 9. “Less than” excludes the edge case of x = 9, but “less than or equal to” includes when x = 9.

Another theorem is that if a number T is less than the smallest element of S, then T is not an element of S. The set minimum theorem puts a rigid ceiling on the set of positive integers that can be denied using that other theorem.

Note that sufficiently large is a special case of “almost all”. In the example given in the first paragraph, the statement is true for all x > 9. For the first 13 integers, the statement is true for 10, 11, 12, and 13; seems not too solid as 4/13 = 30.769%, which is less than a third. But if we extend our number line to cover the first 1000 natural numbers, the statement is true for 991 of those numbers, or 99.1% of the first 1000 natural numbers. If we go to a million, there are 999,991 valid solutions or 99.9991%. You get the point. The set of initial exceptions remains the same no matter how far we go.