Problem 23 - Discrete Mathematics with Applications

Question

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

Step-by-Step Solution

Answer

The short answer based on the given step-by-step solution is: \(A \times (B \cap C) = A \times \{x\} = \{(b,x), (c,x)\}\).

Step 1: Identify the intersection of sets B and C

To find the intersection of sets B and C, we need to identify common elements between both sets. Set \(B\) contains the element \(x\), and set \(C\) contains elements \(x\) and \(z\). The only common element between both sets is \(x\). Therefore, the intersection of sets \(B\) and \(C\) is a set that contains only \(x\): \((B \cap C) = \{x\}\).

Step 2: Find the Cartesian product of sets A and (B ∩ C)

Now that we have the intersection of sets \(B\) and \(C\), we can proceed to find the Cartesian product of sets \(A\) and \(\{x\}\). The Cartesian product is the set of all ordered pairs \((a,b)\) where \(a\) belongs to set \(A\) and \(b\) belongs to the intersection of sets \(B\) and \(C\). So, for this case, we have: \[ A \times (B \cap C) = A \times \{x\} = \{(b,x), (c,x)\} \] Thus, the Cartesian product of sets \(A\) and \(B \cap C\) is the set \(\{(b,x), (c,x)\}\).

Key Concepts

Set Theory and Its Fundamentals

Set theory is a foundational system in mathematics, dealing with the concept of a 'set,' which is simply a collection of distinct objects, considered as an object in its own right. Sets are often denoted by curly braces, for instance, if we consider the set of colors of a rainbow, we could represent it as \(\{red, orange, yellow, green, blue, indigo, violet\}\).

In the realm of set theory, not only can we talk about individual sets, but we can also discuss the relationship between multiple sets through operations like intersections, unions, and the Cartesian product. The intersection of two sets, denoted by \(\cap\), contains all elements that are common to both sets. If there are no common elements, the intersection is an empty set, represented by \(\emptyset\).

Understanding these basics is crucial for tackling more complex set theory concepts. As you progress through set problems, keep these principles in mind: a set is a collection of unique elements, the intersection finds common ground, and the set operations help us derive new sets from existing ones.

Ordered Pairs and Their Significance

At the heart of the Cartesian product is the notion of an 'ordered pair'. An ordered pair, written as \(\left(a, b\right)\), is a fundamental component in mathematics used to couple two elements together where the order of these elements is significant. This concept is crucial when mapping relationships between sets, as the first element \(a\) comes from one set, and the second element \(b\) comes from another.

Unlike sets, where the order of elements does not matter and repeats are not counted, every element in an ordered pair is distinct and position-sensitive. Therefore, \(\left(a, b\right)\) and \(\left(b, a\right)\) are considered different ordered pairs unless \(a = b\). This definition is pivotal when creating graphical representations on a Cartesian plane or even organizing data into a sequence, making ordered pairs a key concept in mathematics and beyond.

Intersection of Sets

The intersection of sets captures the idea of commonality between collections. When we have two or more sets, the intersection is the set containing all elements that are present in each of the sets we're considering. This is symbolized by \(\cap\).

For example, if we take Set \(B\) with \(\{x\}\) and Set \(C\) with \(\{x, z\}\), the intersection of \(B \text{ and } C\), denoted as \(B \cap C\), would be \(\{x\}\), since \(x\) is the only element present in both sets. If there were no common elements, the sets would be described as disjoint, and their intersection would be the empty set \(\emptyset\). This concept of intersection is vital in many areas of mathematics and is frequently used to solve problems involving multiple sets, such as in probability, statistics, and logic. It allows us to focus on the shared attributes of different sets and, by intersection, we can filter elements that satisfy a particular criteria shared by all sets involved.