larger! What about 0? If we divide 0 by 10, we end up with 0 again, so 0 does not “qualify” either. We want
the number to get strictly smaller! Now, addition to the above statement, we can claim its reverse:
(x/10<x)⇒ (x> 0)
This means that if we divide a real number by 10 and get a strictly smaller number, the original number
must be positive.
Logical equivalence
When a logical implication works in both directions, we have logical equivalence. This
means that the left-hand part of the statement is true if and only if the right-hand part is
true. The antecedent can also be the consequent, and vice-versa. To symbolize logical equiva-
lence, we use a double-shafted, double-headed arrow, often with extra space on either side (⇔).
We can also write the cryptic word “iff.” Now we can answer the challenge above with a single
statement. We can write either
(x/10<x)⇔ (x> 0)
or
(x/10<x) iff (x> 0)
How Inequalities Behave
Imagine three variables, a, b, and c.Now suppose that we think of some way to compare
their values. Such a “comparison tool” is called a relation. In algebra, the variables represent
numbers. But in general mathematics, they often represent other things, such as sets or logical
statements.
Three properties of relations
Suppose we symbolize a newly dreamed-up relation by a pound sign (#, read as “pound”).
Let’s define three properties that may (or may not) apply to this relation. Our relation is reflex-
ive if and only if, for all possible values of a,
a # a
Our relation is symmetric if and only if, for all possible values of a and b,
(a # b)⇒ (b # a)
Our relation is transitive if and only if, for all possible values of a, b, andc,
[(a # b) & (b # c)]⇒ (a # c)
How Inequalities Behave 179
