Question

For parts (a) and (b), let \(U\) be any set, and let \(X=P(U)\). (a) Prove that \(X\) with the operations \(\cap\) for meet and \(\cup\) for join is a distributive lattice. (b) Prove that \(X\) with the operations \(\cup\) for meet and \(\cap\) for join is a distributive lattice.

Short answer

Both configurations create distributive lattices.

Step-by-step solution

Define the Power Set and Its Operations

Let \( U \) be a set and \( X = P(U) \), the power set of \( U \). Recall that \( P(U) \) is the set of all subsets of \( U \). The operations \( \cap \) (intersection) and \( \cup \) (union) are defined for the subsets in the power set. In a lattice, 'meet' and 'join' refer to operations analogous to 'greatest lower bound' and 'least upper bound', respectively.

Prove that \( (P(U), \cap, \cup) \) is a distributive lattice

To prove this, we need to show that \( \cap \) distributes over \( \cup \) and \( \cup \) distributes over \( \cap \). Specifically:1. \( A \cap (B \cup C) = (A \cap B) \cup (A \cap C) \): This holds because intersection distributes over union in set theory.2. \( A \cup (B \cap C) = (A \cup B) \cap (A \cup C) \): This also holds because union distributes over intersection in set theory.Both distribution laws are satisfied, proving \( P(U) \) with intersection as meet and union as join is a distributive lattice.

Prove that \( (P(U), \cup, \cap) \) is a distributive lattice

Now, consider union as the meet operation and intersection as the join operation. We need to establish similar distribution properties as before:1. \( A \cup (B \cap C) = (A \cup B) \cap (A \cup C) \): This rephrases the previous distribution, but is structurally the same.2. \( A \cap (B \cup C) = (A \cap B) \cup (A \cap C) \): Similarly, this is simply a restatement of distribution in set theory.Since both conditions hold, \( P(U) \) with union as meet and intersection as join is also a distributive lattice.

Key concepts

Set Theory

Set theory is a branch of mathematical logic that deals with sets, which are collections of objects. Sets can have various types of elements, from numbers to more abstract things like functions or even other sets. The fundamental insight of set theory is that any logical proposition or relationship can be expressed in terms of set membership and operations on sets.

A few key terms that often arise in set theory include:

• Elements: The individual objects contained within a set.
• Subset: A set whose elements are all contained within another set.
• Universal Set (U): The set that contains all objects under consideration, anything outside it is "not part of the universe."

To understand the problem in a distributive lattice context, remember that we need to know how these sets interact through operations such as intersection and union.

Set theory forms the foundational language for much of mathematics, including subjects like topology, algebra, and discrete mathematics.

Power Set

The power set of any given set is a set of all possible subsets of that set, including the empty set and the set itself. If a set \( U \) contains 'n' elements, its power set \( P(U) \) will contain \( 2^n \) elements.

For example, if \( U = \{a, b\} \), then the power set \( P(U) \) includes \( \{\emptyset, \{a\}, \{b\}, \{a, b\}\} \). Each element of \( P(U) \) is a subset of \( U \). This concept is fundamental because the power set allows us to explore all ways to group the items of a set.

Understanding the power set helps in building and analyzing complex sets and their relationships, especially through intersection and union operations. It's a vital concept when dealing with problems in lattice theory, among other areas.

• Power Set Symbol: Denoted as \( P(U) \), where U is any given set.
• Size: Has \( 2^n \) elements if the original set has 'n' elements.

Union and Intersection Operations

Union and intersection are fundamental operations in set theory, allowing us to combine or compare sets. The union of two sets \( A \) and \( B \) is a set containing all elements from both \( A \) and \( B \). It's denoted by \( A \cup B \). Union is akin to finding the "least upper bound" in lattice terminology, including elements of both sets without duplication.

Conversely, the intersection of sets \( A \) and \( B \) is a set containing only the elements that are present in both \( A \) and \( B \). This is symbolized as \( A \cap B \) and parallels the concept of "greatest lower bound" in a lattice, capturing only the shared elements.

• Union (\( A \cup B \)): Combines all elements from both sets.
• Intersection (\( A \cap B \)): Contains only elements common to both sets.

Understanding these operations is crucial when working with set algebra, as they provide the tools to create and manipulate sets within various mathematical structures, including distributive lattices.