1.3: Cartesian Products and Power Sets
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While the last section discussed combining sets of individual members to create another set of individual members, here we discuss creating sets of non-repeating tuples (pairs, triplets, and higher groupings). Later in the course, we will see how to calculate the number of tuples that would be created under various circumstances.
1.3 Cartesian Products and Power Sets
Definition 1.3.1: Cartesian Products
Let A and B be sets. The Cartesian product of A and B, denoted by \(A\ times B\), is defined as follows: \(AxB=\{(a,b)|a \text{ } \epsilon A \text{ } and \text{ } b \epsilon B\}\), that is, \(A \times B\) is the set of all possible ordered pairs whose first component comes from A and whose second component comes from B
Example 1.3.2: Some Cartesian Products
Notation in mathematics is often developed for good reason. In this case, a few examples will make clear why the symbol \(\times\) is used for Cartesian products.
Solution
- Let \(A = \{1, 2, 3\}\) and \(B = \{4, 5\}\). Then \(A×B = \{(1, 4), (1, 5), (2, 4), (2, 5), (3, 4), (3, 5)\}\). Note that \(|A × B| = 6 = |A| × |B|\).
- \(A = \{(1, 1), (1, 2), (1, 3), (2, 1) , (2, 2), (2, 3), (3, 1), (3, 2), (3 , 3) \}\). Note that \(|A \times A| = 9 = |A\^2\).
These two examples illustrate the general rule that if A and B are finite sets, then \(|A \times B| = |A| \times |B|\).
We can define the Cartesian product of three (or more) sets similarly. For example, \(A \times B \times C = \{(a, b, c) : a \text{ } \epsilon A, b \text{ } \epsilon B, c \text{ } \epsilon C \}\).
It is common to use exponents if the sets in a Cartesian product are the same:
\(A^2=A \times A\)
\(A^3= A \times A \times A\)
and in general,
\(A^n=\frac{A \times A\times...\times A}{\text{n factors}}\)
1.3.2 Power Sets
Definition 1.3.3: Power Set
Definition 1.3.3: Power Set. If A is any set, the power set of A is the set of all subsets of A, denoted \(\wp (A)\). The two extreme cases, the empty set and all of A, are both included in \(\wp (A)\).
Example 1.3.4: Some Power Sets
- \(\wp (\emptyset)=\{\emptyset\}\)
- \(\wp (\{1\})=\{\emptyset , \{1\}\}\)
- \(\wp (\{1,2\})=\{\emptyset , \{1\}, \{2\}, \{1,2\}\}\)
We will leave it to you to guess at a general formula for the number of elements in the power set of a finite set.