Question

Find \(A \times B \cup C \backslash B\) where \(A=\\{1,2\\}, B=\\{3,4\\}\), and \(\mathrm{C}=\\{1,2,3,4\\} .\)

Short answer

The short answer based on the step-by-step solution is: \[A \times B \cup C \backslash B = \{(1,3),(1,4),(2,3),(2,4), 1, 2\}\]

Step-by-step solution

Find the Cartesian product of A and B (\(A \times B\))

To find the Cartesian product of \(A \times B\), form all possible ordered pairs where the first element is from set A and the second element is from set B. In this case, the sets are \(A=\\{1,2\\}\) and \(B=\\{3,4\\}\). \[A \times B = \{(1,3), (1,4), (2,3), (2,4)\}\]

Find the set difference of C and B (\(C \backslash B\))

To find the set difference of \(C \backslash B\), remove all elements of set B from set C. In this case, the sets are \(C=\\{1,2,3,4\\}\) and \(B=\\{3,4\\}\). \[C \backslash B = \{1, 2\}\]

Find the union of \(A \times B\) and \(C \backslash B\) (\(A \times B \cup (C \backslash B)\))

To find the union of the sets \(A \times B\) and \(C \backslash B\), combine all elements from both sets without duplicates. The sets are: \[A \times B = \{(1,3), (1,4), (2,3), (2,4)\}\] \[C \backslash B = \{1, 2\}\] \[(A \times B) \cup (C \backslash B) = \{(1,3),(1,4),(2,3),(2,4), 1, 2\}\] Thus, the final answer is: \[A \times B \cup C \backslash B = \{(1,3),(1,4),(2,3),(2,4), 1, 2\}\]

Key concepts

Cartesian Product

A Cartesian Product is a fundamental concept in set theory. It involves creating pairs from two sets. Each pair consists of one element from each set.
For example, when finding the Cartesian Product of sets \(A = \{1, 2\}\) and \(B = \{3, 4\}\), we are creating all possible ordered pairs where the first element is from \(A\) and the second is from \(B\).
The result is:

• \(A \, \times \, B = \{(1,3), (1,4), (2,3), (2,4)\}\)

To visualize, think of it as matching every element of \(A\) with every element of \(B\), like creating a grid. It's important to note that the order in each pair matters, so \((1,3)\) is different from \((3,1)\). This intricacy makes Cartesian Products a valuable tool in data organization and computer science.

Set Difference

Set Difference is a straightforward operation in set theory. It involves subtracting one set from another. Specifically, in the context \(C \setminus B\), this means removing all elements found in \(B\) from \(C\).
Consider \(C = \{1, 2, 3, 4\}\) and \(B = \{3, 4\}\). The difference \(C \setminus B\) is the set of elements that are in \(C\) but not in \(B\).
Let's write it down:

• \(C \setminus B = \{1, 2\}\)

This operation is essential for distinguishing unique elements in one set that are absent in another. It's widely used in logical operations and database queries to filter out unwanted data.

Set Union

Set Union is about combining different sets. This operation gathers all unique elements present in any of the involved sets.
For example, when we have \(A \times B\) from the Cartesian Product and \(C \setminus B\) from the set difference, the union combines them without repetition of elements.

• \((A \times B) \cup (C \setminus B) = \{(1,3),(1,4),(2,3),(2,4),1,2\}\)
Set Union ensures we never double-count elements. It is greatly beneficial anytime we need a comprehensive list of elements from diverse sources, such as in mathematics, statistics, and computer programming.