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190 Relations
be thought of as saying that, in some sense, y is “at least as large as” x.You
might say that each of these statements specifies what order x and y come in.
This motivates the following definition.

Definition 4.4.2. Suppose R is a relation on a set A. Then R is called a partial
order on A (or just a partial order if A is clear from context) if it is reflexive,
transitive, and antisymmetric. It is called a total order on A (or just a total
order) if it is a partial order, and in addition it has the following property:

∀x ∈ A∀y ∈ A(xRy ∨ yRx).
The relations L and S just considered are both partial orders. S is not a total
order, because it is not true that ∀x ∈ B∀y ∈ B(x ⊆ y ∨ y ⊆ x). For example,
if we let x ={1} and y ={2}, then x ⊆ y and y ⊆ x. Thus, although we can
think of the relation S as indicating a sense in which one element of B might
be at least as large as another, it does not give us a way of comparing every
pair of elements of B. For some pairs, such as {1} and {2}, S doesn’t pick out
either one as being at least as large as the other. This is the sense in which the
ordering is partial. On the other hand, L is a total order, because if x and y are
any two real numbers, then either x ≤ y or y ≤ x. Thus, L does give us a way
of comparing any two real numbers.

Example 4.4.3. Which of the following relations are partial orders? Which
are total orders?

1. Let A be any set, and let B = P (A) and S ={(x, y) ∈ B × B | x ⊆ y}.
2. Let A ={1, 2} and B = P (A) as before. Let R ={(x, y) ∈ B × B | y has
at least as many elements as x}= {(∅, ∅), (∅, {1}), (∅, {2}), (∅, {1, 2}),
({1}, {1}), ({1}, {2}), ({1}, {1, 2}), ({2}, {1}), ({2}, {2}), ({2}, {1, 2}), ({1, 2},
{1, 2})}.
3. D ={(x, y) ∈ Z × Z | x divides y}.
+
+
4. G ={(x, y) ∈ R × R | x ≥ y}.
Solutions
1. This is just a generalization of one of the examples discussed earlier, and
it is easy to check that it is a partial order. As long as A has at least two
elements, it will not be a total order. To see why, just note that if a and b
are distinct elements of A, then {a} and {b} are elements of B for which
{a} ⊆{b} and {b} ⊆{a}.
2. Note that ({1}, {2}) ∈ R and ({2}, {1}) ∈ R, but of course {1} ={2}. Thus, R
is not antisymmetric, so it is not a partial order. Although R was defined by