Algebra Know-It-ALL

180 Equations and Inequalities

We can use parentheses, braces, and brackets in logical statements, just as we use them in ordi-

nary mathematical expressions and equations. The ampersand (&) stands for “and.”

If we have a relation # that is reflexive, symmetric, and transitive, then # is called an

equivalence relation. The converse of this is also true: If # is an equivalence relation, then # is

reflexive, symmetric, and transitive.

Behavior of the = relation

You may be thinking, “I’ve seen some of these properties before!” In the solution to Prob. 8

in Chap. 6, we saw that the equality relation (=) is reflexive, symmetric, and transitive, so

it is an equivalence relation. In contrast, the various forms of inequality are not equivalence

relations.

Behavior of the ≠ relation

“Not equal” fails the test when it comes to the reflexive property. This is trivial; if “not equal”

were reflexive, all variables would be different from themselves! The symmetric property, how-

ever, does work for the “not equal” relation. If a is not equal to b, then b is not equal to a.

Logically, we can write this as

(a≠b)⇒ (b≠a)

“Not equal” can’t pass the transitive property test. Suppose we let a be equal to 1, b be equal

to 2, and c be equal to 1. Then we have 1 ≠ 2 and 2 ≠ 1, but it is not true that 1 ≠ 1!

Behavior of the > relation

The “strictly larger” relation is neither reflexive nor symmetric. There is no number a such

thata>a. If a>b, we can never say that b>a, no matter what a and b are. Therefore, “strictly

larger” is not an equivalence relation.

The transitive property does work here. For all numbers a, b, andc, if a>b and b>c,

thena>c. This principle is illustrated in Fig. 11-1.

Mathematicians symbolize the words “for all” by an upside-down capital letter A (∀), and

give it the fancy name universal quantifier. We can now write

(∀a,b,c) : [(a>b) & (b>c)]⇒ (a>c)

The colon separates the quantifier from the main substance of the statement. We can read the

above string of symbols out loud as, “For all a, b, and c: If a is strictly larger than b, and b is

strictly larger than c, then a is strictly larger than c.”

a > b > c

Figure 11-1 The “strictly larger” >

relation is transitive.

If a>b and b>c,

thena>c.