Problem 23 - Discrete Mathematics with Applications

Question

Let \(A=\\{b, c\\}, B=\\{x\\},\) and \(C=\\{x, z\\} .\) Find each set. $$ A \times(B \cup C) $$

Step-by-Step Solution

Answer

The short answer is: \(A \times (B \cup C) = \{(b, x), (b, z), (c, x), (c, z)\}\).

Step 1: Find the union between sets B and C

To find the union between sets B and C, we gather all distinct elements from both sets. Union is represented by the symbol "\(\cup\)". So, we need to find \(B \cup C\). \(B = \{x\}\) and \(C = \{x, z\}\) Now, combine all distinct elements: \(B \cup C = \{x, z\}\)

Step 2: Find the Cartesian product

Now that we have the union of B and C, we can find the Cartesian product of A and the union between B and C. The Cartesian product of two sets is denoted as \(A \times (B \cup C)\) and is equal to the set of all ordered pairs (a, b) where a is in set A and b is in set \(B \cup C\). Set A = \{b, c\} and the union we found earlier was \(B \cup C = \{x, z\}\) Now, let's find the Cartesian product: \(A \times (B \cup C) = \{(a, b) | a\in A, b \in (B \cup C)\} \) The ordered pairs we get are: \((b, x), (b, z), (c, x), (c, z)\) So, the Cartesian product of \(A \times (B \cup C)\) is: \((A \times (B \cup C) = \{(b, x), (b, z), (c, x), (c, z) \}\)

Key Concepts

Union of Sets

Union of sets is a fundamental operation in set theory. It involves combining all the distinct elements from two or more sets. The union is represented using the symbol \(\cup\).

When performing a union, imagine collecting items from each set and creating a new set with all those unique items. For example, given two sets \(B = \{x\}\) and \(C = \{x, z\}\), the union \(B \cup C\) will include all the elements that are in either one or both sets. In this case, \(x\) appears in both, but no second copy is made in the union.

• To find the union, identify all elements in each involved set.
• List each element only once, even if it appears in multiple sets.
• Combine them into a new set, this is your union.

The importance of union comes in merging datasets, maintaining unique entries. It's like filing papers into one folder without duplicates!

Cartesian Product

The Cartesian product is another crucial concept in set theory. It deals with pairing elements from different sets to form ordered pairs. When we say Cartesian product of sets \(A\) and \(B\), denoted as \(A \times B\), we mean forming all possible pairs where the first element is from \(A\) and the second is from \(B\).

This concept creates a plethora of combinations that might be used for different purposes, such as defining functions or relations among sets. For instance, if \(A = \{b, c\}\) and \(B = \{x, z\}\), the Cartesian product \(A \times B\) results in the set of all possible ordered pairs like \((b, x), (b, z), (c, x), (c, z)\).

• The Cartesian product involves taking each element from the first set and pairing it with each from the second set.
• It results in a new set made up of ordered pairs.
• The size of the result set equals the product of the sizes of the individual sets.

By understanding Cartesian products, you get better insights into how associations and relations can be established between different sets.

Ordered Pairs

Ordered pairs represent specific kinds of relationships between two elements, where the order in which they appear is crucial. An ordered pair is written in the form \((a, b)\), and signifies that "\(a\)" comes first, followed by "\(b\)".

This concept is pivotal in set theory and Cartesian products, as it defines relationships clearly by maintaining a specific sequence between elements. For instance, \((b, x)\) and \((x, b)\) are entirely different because the order differs.

• An ordered pair is defined as \((a, b)\), where \(a\) is the first component and \(b\) is the second.
• The pair \((a, b)\) is distinct from \((b, a)\) unless \(a = b\).
• Ordered pairs are essential in mapping elements between sets, creating functions, and defining Cartesian coordinates.

Understanding ordered pairs allows one to appreciate their role in more complex mathematical structures such as graphs, relations, and configurations.