Download Lecture 5 MATH1904 • Disjoint union If the sets A and B have no

Lecture 5 MATH1904
• Disjoint union
If the sets A and B have no elements in
common, we say that they are disjoint
and we write A ∩ B = ∅. If A and B are
disjoint sets, then
|A ∪ B| = |A| + |B|.
• The Addition Principle
If one thing can be selected in a ways
and another thing can be selected in b
ways, then the number of different ways
of selecting the first thing or the second
thing is a + b.
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• Union and intersection
If the sets A and B are not disjoint,
then we have:
|A ∪ B| = |A| + |B| − |A ∩ B|.
The reason is, in counting the sum
|A| + |B|, we have counted the elements
common to A and B twice and so we
need to subtract |A ∩ B| from this sum
to yield |A ∪ B|.
• Cartesian Product
In general, given sets A and B, their
Cartesian product is the set A × B of all
ordered pairs (x, y), where x is an
element of A and y is an element of B.
That is,
A × B = { z | z = (x, y), x ∈ A and y ∈ B }.
The size of A × B is |A| × |B|. This can
be written
|A × B| = |A| × |B|.
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• The Multiplication Principle
If one thing can be selected in a ways
and another thing can be selected in b
ways, then the number of different ways
of selecting the first and the second
thing is ab.
This principle actually goes beyond the
formula for |A × B| because the set from
which the second choice is made could
depend on the first choice.
• Logic
Notice that in the two principles given in
this chapter, addition corresponds to
exclusive or and multiplication
corresponds to and then.
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• Function Definition
A function f : A → B is a subset of
A × B with the property that for every
element x ∈ A there is exactly one
element y ∈ B such that (x, y) ∈ f .
Actually, we’ve seen this before — the
pairs (x, y) correspond to the arrows in
the arrow diagram for f .
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Theory of the Multiplication Principle
The Multiplication Principle can be
described in the language of set theory.
What is perhaps surprising is that this
apparently more general principle can be
obtained from the Addition Principle.
To see this, suppose that we have a set A
and that for each element x ∈ A we have a
set Bx which depends on x. We want to
find the size of the set of ordered pairs
X = { (x, y) | x ∈ A and y ∈ Bx }.
Notice that the sets Bx can depend on
x ∈ A and therefore X is generally not a
Cartesian product (unless there is a set B
such that Bx = B for all x ∈ A.)
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On the other hand, for fixed x ∈ A, the
ordered pairs (x, y), where y ∈ Bx form the
Cartesian product {x} × Bx. Therefore
X=
[
({x} × Bx)
x∈A
and furthermore, the sets {x} × Bx are
disjoint. There is also a one-to-one
correspondence between {x} × Bx and Bx in
which (x, y) corresponds to y. Thus
|{x} × Bx| = |Bx| and, from the Addition
Principle, we have
|X| =
X
|Bx|.
x∈A
In a great many of the examples that we
deal with in this book the sets Bx turn out
to have the same size, say b. When this
happens, we can write |Bx| = b, for all
x ∈ A. If |A| = a, then the formula becomes
|X| = ab.
In fact, this is just the Multiplication
Principle. The special case when all the Bx
are the same set corresponds to the formula
for the size of a Cartesian product.
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