Page 178 - HOW TO PROVE IT: A Structured Approach, Second Edition

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164 Relations
discussing assignments of values to the variables x and y in this statement
are pairs in which the first coordinate is a person and the second is a natural
number. For example, the assignment (Prince Charles, 2) makes the statement
C(x, y) come out true, because Prince Charles does have two children, whereas
the assignment (Johnny Carson, 37) makes the statement false. Note that the
assignment (2, Prince Charles) makes no sense, because it would lead to the
nonsensical statement “2 has Prince Charles children.”
In general, if P(x, y) is a statement in which x ranges over some set A and
y ranges over a set B, then the only assignments of values to x and y that will
make sense in P(x, y) will be ordered pairs in which the first coordinate is an
element of A and the second comes from B. We therefore make the following
definition:
Definition 4.1.1. Suppose A and B are sets. Then the Cartesian product of
A and B, denoted A × B, is the set of all ordered pairs in which the first coor-
dinate is an element of A and the second is an element of B. In other words,
A × B ={(a, b) | a ∈ A and b ∈ B}.

Example 4.1.2.
1. If A ={red, green} and B ={2, 3, 5} then A × B ={(red, 2), (red, 3),
(red, 5), (green, 2), (green, 3), (green, 5)}.
2. If P = the set of all people then P × N ={(p, n) | p is a person and n
is a natural number} = {(Prince Charles, 0), (Prince Charles, 1), (Prince
Charles, 2), . . . , (Johnny Carson, 0), (Johnny Carson, 1), ...}. These are the
ordered pairs that make sense as assignments of values to the free variables
x and y in the statement C(x, y).
3. R × R ={(x, y) | x and y are real numbers}. These are the coordinates of all
2
the points in the plane. For obvious reasons, this set is sometimes written R .
The introduction of a new mathematical concept gives us an opportunity
to practice our proof-writing techniques by proving some basic properties of
the new concept. Here’s a theorem giving some basic properties of Cartesian
products.

Theorem 4.1.3. Suppose A, B, C, and D are sets.
1. A × (B ∩ C) = (A × B) ∩ (A × C).
2. A × (B ∪ C) = (A × B) ∪ (A × C).
3. (A × B) ∩ (C × D) = (A ∩ C) × (B ∩ D).
4. (A × B) ∪ (C × D) ⊆ (A ∪ C) × (B ∪ D).
5. A×∅= ∅×A = ∅.