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6. Relations
When we were talking about logic earlier we considered predicates. These were rules (or
functions) that took one or more objects and returned a value of true or false depending
on whether or not the object(s) had some property or were related in some fashion.
Consider, for example, the "Is-the capital-of" predicate.
true

false
"Is-the-capital-of"(x, y) =
if x is the capital of y
if x is not the capital of y
For example,
"Is-the-capital-of"(Lansing, Michigan) = true
"Is-the-capital-of"(Detroit, Michigan) = false
When we talked about predicates they were used in forming statements that could be
classified as true or false. In this chapter we are going to be looking at predicates again.
However, now they are called relations, and we shall be looking at two special types of
relations, namely equivalence relations and partial orderings.
6.1 Equivalence Relations
In this section we shall be looking at equivalence relations. These are generalizations of
equality. An equivalence relation determines whether two objects are the same in some
respect or have some common property. Often this means that one object can be
substituted for the other in various situations.
Before giving a precise definition of an equivalence relation, let's look at some examples
of relations that are equivalence relations.
p
Example 1. Fractions. A fraction q is the ratio of two integers with a non-zero
1 2
3
denominator q. For example, 2, 4 and 4 are typical fractions. Note that we would
1
2
usually write 2 = 4. What we mean is that we think of them as representing the same
1
number. Technically speaking they are different objects. For example, 2 might represent
2
cutting a pie in two equal pieces and taking one while 4 might represent cutting a pie in
four equal pieces and taking two. The fact that we end up with the same amount of pie in
1
2
p
n
both cases is why we often think of 2 and 4 as the same. In general two fractions q and m
represent the same number if pm = qm.
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The reason for this is the following. Consider the following four scenarios.
i.
Think of cutting a pie into q pieces and taking p of them. This is represented by
p
the fraction q.
ii. Now take an identical pie and cut it into qm pieces and take pm pieces. This is
pm
represented by the fraction qm. We get the same amount of pie as in the first case.
p
pm
So q should be equal to qm.
iii. Again, take an identical pie and cut it into m pieces and taking n of them. This is
n
represented by the fraction m.
iv. Finally, take an identical pie and cut it into qm pieces and take qn pieces. This is
qn
represented by the fraction qm. We get the same amount of pie as in the third case.
n
qn
So m should be equal to qm.
Now compare the the second and fourth cases. In both of these cases we have cut the pie
into qm pieces. We will get the same amount of pie if pm = qn. So we are led to the
p
n
conclusion that q and m represent the same number if pm = qn.
p
n
p
n
We write q = m if pm = qn, although we don't mean q and m are the same object. For
2
3
example, we write 4 = 6.
Example 2. Equivalent algebraic expressions. In high school algebra one does a lot of
work with equivalent algebraic expressions. For example, (x + 1)(x + 2) and x2 + 3x + 2
are equivalent algebraic expressions and we usually write (x + 1)(x + 2) = x2 + 3x + 2.
This does not mean they are the same. It only means that we get the same result if we
substitute the same number into each. For example, if we substitute x = 3 into each we
get (3 + 1)(3 + 2) = (4)(5) = 20 and 32 + (3)(3) + 2 = 9 + 9 + 2 = 20 which are the same.
One place where one substitutes an expression which is equivalent for another is when
one is solving equations. For example, if we are solving the equation x2 + 3x + 2 = 2 the
we can substitute (x + 1)(x + 2) for x2 + 3x + 2 to get (x + 1)(x + 2) = 0 and then conclude
that x = - 1 or x = - 2.
Example 3. Equivalent logical expressions. A
logical expression is a combination of logical
variables (e.g. p, q, etc), logical operations (e.g.
and, or, not, etc.), logical constants (0 and 1) and
parentheses. For example, (pq)', p' + q' and
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p
q
pq
(pq)'
p'
q'
p' + q'
0
0
0
1
1
1
1
0
1
0
1
1
0
1
1
0
0
1
0
1
1
1
1
1
0
0
0
0
p  q are logical expressions. Two logical expressions are equivalent if they have the
same truth table. For example, (pq)' and p' + q' are equivalent since their truth tables are
the same. We have been writing (pq)' = p' + q' to indicate that they are equivalent. One
place where one substitutes an expression which is logically equivalent to another is
when one is designing a logic circuit where all we care about is the logical value of the
output. If one part of the circuit is represented by p' + q' then we can substitute a NAND
gate (pq)'.
Example 4. Numbers congruent mod 12. Earlier we have seen several situations
where we mod the result of an arithmetic calculation involving additions, subtractions
and multiplications by some number p to get the final result. In this case, during the
calculation we can replace any number by another number that has the same remainder
when we divide by p. In this context the two numbers are equivalent.
To make this concrete, let's suppose p = 12. Thus we are working with numbers on a
clock. We say two integers n and m are equivalent mod 12 if they have the same
remainder when we divide by 12, i.e. n mod 12 = m mod 12. Let's write n  m if this is
the case. For example 15  27 since both have a remainder 3 when we divide by 12. In
terms of clocks, the clock reads the same if 15 hours pass as if 27 hours pass. However,
15 / 29 since 15 has remainder 3 when we divide by 12 and 29 has a remainder 5 when
we divide by 12. The clock does not read the same if 15 hours pass as if 29 hours pass.
Equivalence mod 12 is another example of an equivalence relation.
Example 5.  for expressions. Earlier when we were comparing algorithms we defined
what it meant for two expressions to be of the same order as n  . If Sn and Tn are two
S
expressions involving n and 0 < lim Tnn <  then they are of the same order as n   and
n
we write Sn = (Tn). For example, n2 + 3n + 2 = (n2), but n  (n2). Being of the same
order is another example of an equivalence relation.
Example 6. Congruence for Triangles. Two triangles ABC and DEF are congruent
if corresponding angles are equal and the lengths of corresponding sides are equal, i.e.
__ __ __ __
__ __
__
A = D, B = E, C = F, AB = DE, BC = EF and CA = FD. Here AB denotes the
length of the line segment joining A and B. Congruence of triangles is another example
of an equivalence relation.
Example 7. Similarity for Triangles. Two triangles ABC and DEF are similar if
corresponding angles are equal and the lengths of pairs of corresponding sides are
__ __ __ __ __ __
proportional, i.e. A = D, B = E, C = F and AB/DE = BC/EF = CA/FD.
Similarity of triangles is another example of an equivalence relation.
Example 8. Same number of elements for sets. In some situations we consider two
sets A and B as equivalent if they have the same number of elements. Let's write A  B if
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A and B have the same number of elements. For example, {a, b, c}  {e, f, g} but
{a, b, c} / {e, f}. This is also an equivalence relation.
Example 8. Have the same two parents. We say two people are siblings in the strong
sense if both parents are the same. This is an equivalence relation.
Example 9. Give the same output for the same input. We say two C++ functions are
equivalent if they return the same value if they are given the same arguments.
People reflecting on examples such as these noticed that all these relations had three
properties in common. Relations that have these three properties are now called
equivalence relations.
Definition 1. A relation is an equivalence relation if it has the following three properties.
i.
It is transitive
ii.
It is symmetric
iii.
It is reflexive
A relation  is transitive if whenever one has both a  b and b  c then one also has a  c.
A relation  is symmetric if whenever one has a  b then one also has b  a. A relation 
is reflexive if a  a for every object a.
Often it is easy to verify that a certain relation has these three properties and is, therefore,
an equivalence relation. In other cases verifying one or the other of these properties may
require some work. This is particularly true of the transitive property. Let's see how
these definitions work for the examples above.
Example 10 - Equivalent logical expressions. Two logical expressions are equivalent
if they have the same value for every combination of values of the variables appearing in
them. Let's show this is transitive. Suppose P, Q and R are logical expression with
P = Q and Q = R. Then P and Q have the same value for every combination of values of
the variables and Q and R also have the same value for every combination of values of
the variables. Therefore, P and R have the same value for every combination of values of
the variables. So P = R. This shows the relation is transitive. Now let's show it is
symmetric. Suppose P = Q. Then P and Q have the same value for every combination of
values of the variables. Therefore, Q and P have the same value for every combination of
values of the variables. So Q = P. This shows the relation is symmetric. Finally to show
reflexive, we must show P = P for any P. This is true since P and P have the same value
for any combination of values of the variables. So the relation is reflexive. So
equivalence of logical expressions meets the three requirements of an equivalence
relation.
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This proof that equivalence of logical expressions is an equivalence relation works with
minor modifications for most of the other relations in the examples above since they say
that two objects bear the relation to each other if something computed from each is the
same. However, with equality of fractions one needs to use a different argument.
p
n
Example 11 – Equality of fractions. In this case q = m means pm = nq. Let's show this
p
n
n
r
is transitive. Suppose q = m and m = s. Then pm = nq and ns = mr. If we multiply the
first by s and the second by q we get pms = nqs and nqs = mqr. So pms = mqr. Dividing
p
r
by m gives ps = qr. So q = s . This shows the relation is transitive. Now let's show it is
p
n
n
p
symmetric. We must show q = m implies m = q . The first implies pm = nq and to show
the second we need nq = pm. However, these are the same so the relation is symmetric.
p
p
Finally to show reflexive, we must show q = q . To show this we must show pq = pq.
Since this is true the relation is reflexive. So equality for fractions meets the three
requirements of an equivalence relation.
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