Set Natural
January 7th, 2011
Use mathematical induction to prove that every nonempty finite set of natural numbers has a least element?
Please help!
Let P(n) be the statement “Any set of n natural numbers has a least element.” for n ≥ 1. Clearly P(1) is true as the set singleton set {x} has x as its least element ( x ≤ y is true for all y in {x}.) Moreover, P(2) since given any subset {x,y} of the natural numbers either x < y or y < x, and the set has a least element.
Suppose that for some k ≥ 2 that P(k) is true and consider the statement P(k+1). Let
S={x1, x2, ...., x_k, x_(k+1)}
be any set of natural numbers. We can remove one element x_i and consider the subset of S
S - {x_i}.
Since k ≥ 2, S - {x_i} is a nonempty subset of the natural with k elements. By the induction hypothesis it has a least element x_j. If x_j < x_i, then x_j is the least element of S. If x_i < x_j, then x_i is the least element of S. So in either case, S has a least element. We can conclude that
P(n) is true for all natural numbers n.
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