Question

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

Short answer

The power set \(P(U)\), with \(\cup\) as meet and \(\cap\) as join, forms a complemented lattice.

Step-by-step solution

Understanding the Question

We are given a set \(U\) and we need to consider \(X = P(U)\), the power set of \(U\). Our task is to prove that \(X\) is a complemented lattice with \(\cup\) (union) as meet and \(\cap\) (intersection) as join.

Definitions and Lattice Structure

A lattice is a partially ordered set \((L, \leq)\) where any two elements have a unique supremum (join \(\vee\)) and infimum (meet \(\wedge\)). Here, the ordering is subset inclusion \(\subseteq\), meet \(\wedge\) is \(\cup\), and join \(\vee\) is \(\cap\).

Verify Lattice Properties

For sets \(A, B \in P(U)\):1. Meet: \( A \cup B \) is a set containing all elements from both sets, satisfying idempotent, commutative, and associative properties.2. Join: \( A \cap B \) is a set of elements common to both sets, also satisfying idempotent, commutative, and associative properties.

Order Structure

In \(P(U)\), \(A \subseteq B\) if \(\forall x, x \in A \Rightarrow x \in B\). This is a partial order where each pair of sets \(A\) and \(B\) has a meet \(A \cup B\) (largest set under both) and join \(A \cap B\) (smallest set containing both).

Complements in the Lattice

A complemented lattice requires every element \(A\) to have a complement \(A^c\) such that:1. \(A \cap A^c = \emptyset\)2. \(A \cup A^c = U\).In \(P(U)\), the complement of \(A\) is \(A^c = U \setminus A\). Both conditions are satisfied, confirming that \(X\) is a complemented lattice.

Conclusion

Since all the lattice properties and the definition of a complemented lattice are verified, \(P(U)\) with \(\cup\) for meet and \(\cap\) for join is indeed a complemented lattice.

Key concepts

Power Set

The concept of a power set is foundational in set theory. A power set, denoted as \( P(S) \), is the set of all possible subsets of a given set \( S \), including the empty set and \( S \) itself.

For example, if we consider a set \( U = \{a, b\} \), the power set \( P(U) \) would be \( \{\emptyset, \{a\}, \{b\}, \{a, b\}\} \). This illustrates that the power set contains every possible combination of elements from \( U \).

In case of large sets, the size of the power set increases exponentially. If a set \( S \) has \( n \) elements, then \( P(S) \) will have \( 2^n \) elements. This is because each element can either be included or excluded from a subset.

The power set concept is particularly important in forming partial orders and dealing with set operations, where understanding all possible subsets of a set provides a powerful tool for logical arguments and mathematical proofs.

Set Operations

Set operations are fundamental tools that allow us to manipulate and compare sets. In this context, operations like union (\( \cup \)) and intersection (\( \cap \)) are crucial when studying the properties of sets.

• **Union (\( A \cup B \))**: This operation combines all elements from sets \( A \) and \( B \). If an element is in \( A \) or \( B \) (or both), it appears in \( A \cup B \).
• **Intersection (\( A \cap B \))**: This operation identifies common elements between sets \( A \) and \( B \). Only those elements that are in both sets will be included in \( A \cap B \).

These operations have properties such as:

• **Idempotent**: Performing the operation on a set with itself yields the set itself (e.g., \( A \cup A = A \)).
• **Commutative**: The order of the sets in the operation doesn't affect the result (e.g., \( A \cup B = B \cup A \)).
• **Associative**: Grouping of operations doesn't change the result (e.g., \( (A \cup B) \cup C = A \cup (B \cup C) \)).

Understanding these operations helps structure the relationships and the calculations involved in determining lattice properties.

Partial Order

A partial order is a mathematical concept used to describe a binary relation over a set that is reflexive, antisymmetric, and transitive. For a set \( A \) and a relation \( \leq \) on \( A \), it fulfills:

• **Reflexivity**: Every element is related to itself (\( a \leq a \) for all \( a \in A \)).
• **Antisymmetry**: If two elements are related in both directions, they must be equal (if \( a \leq b \) and \( b \leq a \), then \( a = b \)).
• **Transitivity**: If one element is related to a second, and the second is related to a third, then the first is related to the third (if \( a \leq b \) and \( b \leq c \), then \( a \leq c \)).

In the context of power sets, this partial order is often the subset relation \( \subseteq \).

This means for any subsets \( A \) and \( B \) within a power set \( P(U) \), \( A \subseteq B \) indicates that all elements of \( A \) are also elements of \( B \).

Such partial orders help in constructing lattices, where they define the hierarchy or structure within a set, facilitating operations like finding meets and joins.