Algebra Know-It-ALL

Behavior of the < relation

The “strictly smaller” relation fails the reflexive and symmetric tests, just as does the “strictly
larger” relation. In fact, we can restate the case here by turning all the inequality symbols
around. There is no number a such that a<a. If a<b, we can never say that b<a,regardless
of the values of a and b. The “strictly smaller” relation is not an equivalence relation.
The “strictly smaller” relation is transitive. For all numbers a, b, and c, if a<b and b<c,
thena<c. This principle is illustrated in Fig. 11-2. We can write

(∀a,b,c) : [(a<b) & (b<c)]⇒ (a<c)

Behavior of the ê relation

The “larger than or equal” relation is not symmetric. If a≥b, thenb≥a if a and b happen
to be the same, but it never works if a is larger than b. Therefore, the “larger than or equal”
relation is not an equivalence relation.
The reflexive and transitive properties hold true for the “larger than or equal” relation.
A number a is always larger than or equal to itself, simply because it equals itself, and that’s
good enough! For all numbers a, b, andc, if a≥b and b≥c, then a≥c. We can write these
two facts in formal terms as

(∀a) : a≥a

and

(∀a,b,c) : [(a≥b) & (b≥c)]⇒ (a≥c)

Behavior of the Ä relation

The case for the “smaller than or equal” relation is similar to the case for the “larger than or
equal” relation. If a≤b, thenb≤a if a=b, but never if a<b. Because of this, the “strictly
smaller” relation cannot qualify as an equivalence relation. The reflexive and transitive proper-
ties, however, do hold true here. We can logically state these fact as

(∀a) : a≤a

and

(∀a,b,c) : [(a≤b) & (b≤c)]⇒ (a≤c)

a > b > c

Figure 11-2 The “strictly smaller” >

relation is transitive.

If a < b and b < c,

thena < c.

How Inequalities Behave 181