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)
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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.
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, then b≤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) b >
a > c
Figure 11-2 The “strictly smaller” >
relation is transitive.
If a < b and b < c, thena < c.
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Are you confused?
Once you know what all these symbols mean, you can read complicated logical statements out loud or write them down in words. You can even break them up and turn them into “mathematical verse.” For example, the last logical sentence in the previous paragraph can be written like this.
For all a, b, and c:
If
a is smaller than or equal to b, and
b is smaller than or equal to c, then
a is smaller than or equal to c.
Here’s a challenge!
Is logical implication an equivalence relation?
Solution
No. To see why, let’s make up three statements and give them logical names:
• I’m thinking of a natural number =Tn.
• I’m thinking of a rational number =Tq.
• I’m thinking of a real number =Tr.
Now let’s check out the symmetric property. From our knowledge of how the sets of naturals, rationals, and reals (N, Q,andR) are related, we know that
Tn⇒Tq and
Tq⇒Tr These statements translate to the following mathematical verses.
If
I’m thinking of a natural number, then
I’m thinking of a rational number.
If
I’m thinking of a rational number, then
I’m thinking of a real number.
We can’t reverse either of these implications and still end up with a valid statement. If I’m thinking of a rational number, I’m not necessarily thinking of a natural number. (Suppose it’s 1/2.) If I’m thinking of a real number, I’m not necessarily thinking of a rational number. (Suppose it’s the positive square root of 2.) Logical implication is not an equivalence relation, because it’s not symmetric.