Question

Prove that (i) \((A \cup B) \times C=(A \times C) \cup(B \times C)\); (ii) \((A \cap B) \times(C \cap D)=(A \times C) \cap(B \times D)\); \((\) iii \()(X \times Y)-\left(X^{\prime} \times Y^{\prime}\right)=\left[\left(X \cap X^{\prime}\right) \times\left(Y-Y^{\prime}\right)\right] \cup\left[\left(X-X^{\prime}\right) \times Y\right]\) [Hint: In each case, show that an ordered pair \((x, y)\) is in the left-hand set iff it is in the right-hand set, treating \((x, y)\) as one element of the Cartesian product.]

Short answer

All statements are proven using set theory logic by verifying membership conditions for each side of the equations.

Step-by-step solution

Prepare the Groundwork

In each part, we'll be dealing with Cartesian products, unions, intersections, and set differences. We need to remember that Cartesian products like \(A \times C\) consist of all ordered pairs \((a, c)\) where \(a \in A\) and \(c \in C\). Similarly, \(A \cup B\) contains all elements in either \(A\) or \(B\), and \(A \cap B\) contains all elements common to both \(A\) and \(B\). Set difference \(X - Y\) consists of elements in \(X\) but not in \(Y\).

Prove Part (i)

We start with \[(A \cup B) \times C = (A \times C) \cup (B \times C)\]Assume \((x, y) \in (A \cup B) \times C\). Then \(x \in A \cup B\) and \(y \in C\). Thus, \(x\) is either in \(A\) or \(B\). If \(x \in A\), then \((x, y) \in A \times C\). If \(x \in B\), then \((x, y) \in B \times C\). Thus, \((x, y) \in (A \times C) \cup (B \times C)\).Conversely, if \((x, y) \in (A \times C) \cup (B \times C)\), then \((x, y) \in A \times C\) or \((x, y) \in B \times C\), which implies \(x \in A\) or \(x \in B\) and \(y \in C\). Hence, \((x, y) \in (A \cup B) \times C\).

Prove Part (ii)

Now, consider \[(A \cap B) \times (C \cap D) = (A \times C) \cap (B \times D)\]Assume \((x, y) \in (A \cap B) \times (C \cap D)\). This means \(x \in A \cap B\) and \(y \in C \cap D\), implying \(x \in A\), \(x \in B\), \(y \in C\), and \(y \in D\). Therefore, \((x, y) \in A \times C\) and \((x, y) \in B \times D\), so \((x, y) \in (A \times C) \cap (B \times D)\).Conversely, if \((x, y) \in (A \times C) \cap (B \times D)\), then \((x, y) \in A \times C\) and \((x, y) \in B \times D\). So, \(x \in A\), \(x \in B\), \(y \in C\), and \(y \in D\), meaning \(x \in A \cap B\) and \(y \in C \cap D\), thus \((x, y) \in (A \cap B) \times (C \cap D)\).

Prove Part (iii)

Finally, we tackle \[(X \times Y) - (X' \times Y') = [(X \cap X') \times (Y - Y')] \cup [(X - X') \times Y]\]Assume \((x, y) \in (X \times Y) - (X' \times Y')\). This means \((x, y) \in X \times Y\) but \((x, y) ot\in X' \times Y'\). Thus \(x \in X\) and \(y \in Y\), but not both \(x \in X'\) and \(y \in Y'\). Either \(x \in X\) and \(y ot\in Y'\), which means \((x, y) \in (X \cap X') \times (Y - Y')\), or \(x ot\in X'\) and \(y \in Y\), which means \((x, y) \in (X - X') \times Y\).Conversely, if \((x, y) \in [(X \cap X') \times (Y - Y')]\), then \(x \in X \cap X'\) and \(y \in Y - Y'\), hence \(x \in X\), \(x ot\in X'\), \(y \in Y\) and \(y ot\in Y'\), meaning \((x, y) \in (X \times Y) - (X' \times Y')\). Similarly, if \((x, y) \in (X - X') \times Y\), it implies \((x, y) \in (X \times Y) - (X' \times Y')\).

Key concepts

Set Theory

Set theory is a branch of mathematical logic that deals with sets, which are collections of objects. It provides the foundation for many areas of mathematics. A set can contain objects, called elements, and is typically described using brackets, for example, \( \{a, b, c\} \). Within any set, there are no repeated elements, as each element is unique.

In set theory, we're not limited to numbers. Sets can be comprised of any type of object, such as numbers, people, or even other sets. Understanding set theory helps us delve into more complex concepts such as relations and functions. It is used to define almost all mathematical objects. Key operations include:

• Union: Combining the elements of two sets.
• Intersection: Finding common elements between sets.
• Difference: Subtracting one set from another.

By using set theory, we can describe complex relationships as simple as checking contents or applying operations like union and intersection. For example, mathematicians use set theory to express logical conclusions about groupings and combinations.

Unions and Intersections

Unions and intersections are common operations in set theory that deal with the combining and comparing of sets.

Unions involve combining all the elements from two or more sets into a new set. For instance, for sets \(A\) and \(B\), the union \(A \cup B\) consists of all elements that are either in \(A\), \(B\), or in both. If \(A = \{1, 2\}\) and \(B = \{2, 3\}\), then \(A \cup B = \{1, 2, 3\}\).

Conversely, intersections concern the elements common to all sets being compared. For the same sets \(A\) and \(B\), the intersection \(A \cap B\) results in a set of elements that occur in both \(A\) and \(B\). Using our earlier example, \(A \cap B = \{2\}\), because 2 is the only element present in both sets.

• Unions are inclusive, incorporating all elements.
• Intersections are exclusive, limiting to common elements.

These operations are crucial for solving problems that involve combining data from different sources or lists and finding commonalities, such as in database queries or probability.

Ordered Pairs

Ordered pairs are a fundamental concept in mathematics, especially in the study of Cartesian products and set theory. An ordered pair is typically written as \((a, b)\), where the order of the elements does matter. It represents a pair of values or objects.

In the Cartesian plane, for example, a point is represented as an ordered pair \((x, y)\), where \(x\) and \(y\) are the coordinates of the point. Unlike sets, which do not consider order, ordered pairs have a defined sequence. \((a, b)\) is not the same as \((b, a)\) unless \(a = b\).

Ordered pairs are essential for defining relations and functions, as they provide a way to connect elements from two different sets. The Cartesian product combines two sets, \(A\) and \(B\), into a set of ordered pairs \(A \times B\). If \(A = \{1, 2\}\) and \(B = \{3, 4\}\), the Cartesian product \(A \times B\) results in \(\{(1, 3), (1, 4), (2, 3), (2, 4)\}\). Understanding how ordered pairs work is crucial for advanced concepts such as vector spaces and coordinate systems.

Set Difference

The concept of set difference is another key operation in set theory, used to find elements that belong to one set but not to another. The set difference between two sets, \(X\) and \(Y\), is denoted as \(X - Y\).

To calculate a set difference, you identify all elements present in the first set that are not in the second. For example, if \(X = \{1, 2, 3\}\) and \(Y = \{2, 3, 4\}\), then \(X - Y = \{1\}\), because 1 is the only element in \(X\) that is not in \(Y\).

• Only elements unique to \(X\) remain in \(X - Y\).
• Mathematically, the expression can be written as \(X - Y = \{x \mid x \in X \text{ and } x otin Y\}\).

Set difference is especially useful for solving problems that require identifying unique elements or when you need to filter data. It's an operation that appears often in applications such as database management, where determining exclusion is necessary. Understanding set differences can aid comprehension in more complex operations involving complement and symmetric differences.