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Chapter 5: Problem 3

Prove Theorem 5.25, Part (1): \(A \times(B \cap C)=(A \times B) \cap(A \times C)\).

Short Answer

Expert verified
To prove that \(A \times (B \cap C) = (A \times B) \cap (A \times C)\), we first showed that \(A \times (B \cap C) \subseteq (A \times B) \cap (A \times C)\) by taking an arbitrary element from \(A \times (B \cap C)\) and demonstrating it must also be in the intersection. Then, we showed the reverse inclusion \((A \times B) \cap (A \times C) \subseteq A \times (B \cap C)\) by taking an arbitrary element from the intersection and demonstrating it must be in \(A \times (B \cap C)\). As both inclusions hold, we conclude that \(A \times (B \cap C) = (A \times B) \cap (A \times C)\), thus proving Theorem 5.25, Part (1).

Step by step solution

01

Show that \(A \times (B \cap C) \subseteq (A \times B) \cap (A \times C)\)

Let's take an arbitrary element in \(A \times (B \cap C)\), say \((a, b_c)\), where \(a \in A\) and \(b_c \in (B \cap C)\). Since \(b_c \in (B \cap C)\), this implies that \(b_c \in B\) and \(b_c \in C\). Consequently, \((a, b_c) \in A \times B\) and \((a, b_c) \in A \times C\). Now, since \((a, b_c)\) is in both \(A \times B\) and \(A \times C\), it must also be in their intersection: \((a, b_c) \in (A \times B) \cap (A \times C)\). As this holds for any arbitrary element of \(A \times (B \cap C)\), we can conclude that \(A \times (B \cap C) \subseteq (A \times B) \cap (A \times C)\).
02

Show that \((A \times B) \cap (A \times C) \subseteq A \times (B \cap C)\)

Now, let's take an arbitrary element in the intersection \((A \times B) \cap (A \times C)\), say \((a, b_c)\). Since \((a, b_c)\) is in the intersection, it must be in both \(A \times B\) and \(A \times C\). This implies that \(b_c \in B\) and \(b_c \in C\), and therefore \(b_c \in (B \cap C)\). Consequently, \((a, b_c) \in A \times (B \cap C)\). As this holds for any arbitrary element of \((A \times B) \cap (A \times C)\), we can conclude that \((A \times B) \cap (A \times C) \subseteq A \times (B \cap C)\).
03

Conclude the Equality

As we have shown in Steps 1 and 2 that \(A \times (B \cap C) \subseteq (A \times B) \cap (A \times C)\) and \((A \times B) \cap (A \times C) \subseteq A \times (B \cap C)\), we can now conclude that \(A \times (B \cap C) = (A \times B) \cap (A \times C)\). This completes the proof of Theorem 5.25, Part (1).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Set Theory
Set theory forms the foundation of mathematics, dealing with the concept of a collection of distinct objects, considered as an entity in its own right. When we talk about sets, we're referring to the grouping of elements that may have a certain property in common or follow a specific rule.

For example, in the provided exercise, the sets designated as \(A\), \(B\), and \(C\) could contain numbers, characters, or any other definable entities, and the operations used on these sets (such as the intersection represented by \(B \cap C\)) adhere to specific rules within set theory.

Understanding the fundamental operations like union (\(\cup\)), intersection (\(\cap\)), and difference, along with the concept of subsets (denoted by \(\subseteq\))), is crucial for establishing a solid base in set theory, which then forms the groundwork for more advanced mathematical concepts.
Cartesian Product
The Cartesian product is an operation that returns a set from all the possible ordered pairs created by the elements of two sets. An ordered pair \( (a, b) \) has a first element \(a\) from the first set and a second element \(b\) from the second set.

In the context of our exercise, \(A \times (B \cap C)\) generates a set of ordered pairs where the first element comes from set \(A\), and the second element is taken from the intersection of sets \(B\) and \(C\). This operation is essential in areas such as coordinate geometry, where we often deal with points defined by pairs of numbers corresponding to axes.
Proof Techniques
Proof techniques are the methods used to establish the truth of mathematical statements. Some commonly used techniques include direct proof, indirect proof (also known as proof by contradiction), and proof by induction.

In the solution given, a direct proof technique has been employed, where we demonstrate a statement by straightforward logical steps. The process involves two subsets, showing first that each element of the first subset is also an element of the second subset (\(\subseteq\)), and then proving the reverse.

The success of such a technique resides in the clarity and logical progression of the argument. Understanding how to structure these proofs is vital for mathematics students, as proof writing is a skill that refines logical thinking and problem-solving abilities.
Theorem Proving
Theorem proving is the act of demonstrating that a statement is true based on accepted axioms and inference rules. In our exercise, Theorem 5.25, Part (1) states that \(A \times (B \cap C) = (A \times B) \cap (A \times C)\). The proof requires showing that the set produced by the Cartesian product of set \(A\) with the intersection of sets \(B\) and \(C\) is the same as the intersection of two separate Cartesian products.

The proof provided for the theorem shows the equality of these two sets by verifying that each is a subset of the other, which concludes that the sets are identical. Understanding how to establish such equalities is a key aspect of theorem proving, which itself is a fundamental part of advancing mathematical theory.

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Most popular questions from this chapter

Let \(C=\\{x \in \mathbb{Z} \mid x \equiv 7(\bmod 9)\\}\) and \(D=\\{x \in \mathbb{Z} \mid x \equiv 1(\bmod 3)\\}\). (a) List at least five different elements of the set \(C\) and at least five elements of the set \(D\). (b) Is \(C \subseteq D\) ? Justify your conclusion with a proof or a counterexample. (c) Is \(D \subseteq C\) ? Justify your conclusion with a proof or a counterexample.

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