Page 206 - HOW TO PROVE IT: A Structured Approach, Second Edition

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192 Relations
Not every set of words will have an element that is smallest in this sense. For
example, consider the set C ={a, me, men, tame, mental}⊆ A. Each of the
words men, tame, and mental is larger than at least one other word in the set,
but neither a nor me is larger than anything else in the set. We’ll call a and me
minimal elements of C. But note that neither a nor me is the smallest element
of C in the sense described in the last paragraph, because neither is smaller
than the other. The set C has two minimal elements but no smallest element.
These examples might raise a number of questions in your mind about small-
est and minimal elements. The set C has two minimal elements, but B has only
one smallest element. Can a set ever have more than one smallest element?
Until we have settled this question, we should only talk about an object being
a smallest element of a set, rather than the smallest element. If a set has only
one minimal element, must it be a smallest element? Can a set have a smallest
element and a minimal element that are different? Would the answers to these
questions be different if we restricted our attention to total orders rather than
all partial orders? Before we try to answer any of these questions, we should
state the definitions of the terms smallest and minimal more carefully.

Definition 4.4.4. Suppose R is a partial order on a set A, B ⊆ A, and b ∈ B.
Then b is called an R-smallest element of B (or just a smallest element if R
is clear from the context) if ∀x ∈ B(bRx). It is called an R-minimal element
(or just a minimal element) if ¬∃x ∈ B(xRb ∧ x = b).

Example 4.4.5.

1. Let L ={(x, y) ∈ R × R | x ≤ y}, as before. Let B ={x ∈ R | x ≥ 7}.
Does B have any L-smallest or L-minimal elements? What about the set
C ={x ∈ R | x > 7}?
2. Let D be the divisibility relation defined in part 3 of Example 4.4.3. Let B =
{3, 4, 5, 6, 7, 8, 9}. Does B have any D-smallest or D-minimal elements?
3. Let S ={(X, Y) ∈ P (N) × P (N) | X ⊆ Y}, which is a partial order on the
set P (N). Let F ={X ∈ P (N) | 2 ∈ X and 3 ∈ X}. Note that the elements
of F are not natural numbers, but sets of natural numbers. For example,
{1, 2, 3} and {n ∈ N | n is prime} are both elements of F. Does F have any
S-smallest or S-mimimal elements? What about the set G ={X ∈ P (N) |
either 2 ∈ X or 3 ∈ X}?

Solutions
1. Clearly 7 ≤ x for every x ∈ B,so ∀x ∈ B(7Lx) and therefore 7 is a smallest
element of B. It is also a minimal element, since nothing in B is smaller