Question

Write the following sets by listing their elements between braces. $$ \mathscr{P}(\\{1,2\\}) \times \mathscr{P}(\\{3\\}) $$

Short answer

\(\mathscr{P}(\{1,2\}) \times \mathscr{P}(\{3\}) = \{(\emptyset,\emptyset), (\emptyset,\{3\}), (\{1\},\emptyset), (\{1\},\{3\}), (\{2\},\emptyset), (\{2\},\{3\}), (\{1,2\},\emptyset), (\{1,2\},\{3\})\}\)

Step-by-step solution

Calculate Power Set of {1,2}

Find all possible subsets of {1,2}. These are {}, {1}, {2} and {1,2}. So, \(\mathscr{P}(\{1,2\}) = \{\emptyset, \{1\}, \{2\}, \{1,2\}\}\)

Calculate Power Set of {3}

Find all possible subsets of {3}. These are {}, {3}. So, \(\mathscr{P}(\{3\}) = \{\emptyset, \{3\}\}\)

Calculate Cartesian Product

Calculate the Cartesian product \(\mathscr{P}(\{1,2\}) \times \mathscr{P}(\{3\})\). This is the set of all ordered pairs (a, b) where 'a' is any subset from \(\mathscr{P}(\{1,2\})\) and 'b' is any subset from \(\mathscr{P}(\{3\})\). The result is \((\emptyset,\emptyset)\), \((\emptyset,\{3\})\), \((\{1\},\emptyset)\), \((\{1\},\{3\})\), \((\{2\},\emptyset)\), \((\{2\},\{3\})\), \((\{1,2\},\emptyset)\), \((\{1,2\},\{3\})\)

Key concepts

Power Set

The concept of a power set is fundamental in the study of set theory. The power set, denoted as \( \mathscr{P}(S) \), of any given set \( S \) is a set containing all possible subsets of \( S \), including the empty set \( \emptyset \) and \( S \) itself. Imagine a set as a box containing elements; the power set includes every possible combination of items you could take out of that box, even taking nothing (the empty set) or everything (the set itself).

For example, considering a set \( \{1, 2\} \), its power set includes \( \emptyset \) (since not taking any element is also a subset), \( \{1\} \) and \( \{2\} \) (taking each element individually), and \( \{1,2\} \) (taking both elements). Thus, \( \mathscr{P}(\{1,2\}) = \{ \emptyset, \{1\}, \{2\}, \{1,2\} \} \).

The power set concept is pivotal when discussing topics like combinatorics, probability, and Boolean algebra, as it provides a comprehensive list of all possible combinations of set elements.

Ordered Pairs

An ordered pair is a fundamental structure in mathematics used to group two objects where the order of the elements is significant. The first element is called the 'x-coordinate', while the second is the 'y-coordinate'. Ordered pairs are often written in the form \( (x, y) \), and are employed across various branches of mathematics, including coordinate geometry and set theory.

In terms of sets, when we speak of ordered pairs, we highlight the importance of arrangement: \( (a, b) \) is not the same as \( (b, a) \) unless \( a = b \). This notion becomes critical when discussing Cartesian products, where the resulting set consists of all possible ordered pairs made from two given sets. For instance, taking the Cartesian product of the power sets \( \mathscr{P}(\{1,2\}) \times \mathscr{P}(\{3\}) \) means you're pairing every element of the first power set with every element of the second power set, leading to a set of tuples reflecting every possible pairing.

Subset

In set theory, the term 'subset' is used when every element of one set, say \( A \), is also an element of another set, denoted \( B \). In such cases, we write \( A \subseteq B \). Every set \( A \) is considered a subset of itself \( (A \subseteq A) \) and every set has at least one subset, the empty set \( \emptyset \).

Subsets form the building blocks for power sets. When we calculate \( \mathscr{P}(S) \), we essentially list every possible subset of \( S \) which includes variations with differing numbers of elements. For a concrete illustration, the set \( \{3\} \) has two subsets: \( \emptyset \) and \( \{3\} \). This knowledge of subsets aids in understanding the composition of power sets, which in turn are used in operations like finding Cartesian products, to showcase the relationships between different sets.