Problem 32 - Discrete Mathematics with Applications

Question

Is \(\langle S, \cup, \cap,, \emptyset, U)\) a boolean algebra for each subset \(S\) of \(P(U),\) where \(U=[a, b, c] ?\) $$ \\{\emptyset,|a|,|b|,\\{a, b |, U\\} $$

Step-by-Step Solution

Answer

The given structure 𝑆 is not a Boolean algebra because it does not satisfy the Complement Law for all subsets of P(U). All other axioms are satisfied.

Step 1: Determine the Power set of U (P(U))

Calculate the possible subsets of the given set U to form its power set. P(U) = {∅, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}}

Step 2: Check the Commutative Law

Verify the Commutative law is true for the subsets of P(U). A ∪ B = B ∪ A and A ∩ B = B ∩ A The Commutative law is satisfied for all subsets of P(U).

Step 3: Check the Associative Law

Verify the Associative law is true for the subsets of P(U). (A ∪ B) ∪ C = A ∪ (B ∪ C) and (A ∩ B) ∩ C = A ∩ (B ∩ C) The Associative law is satisfied for all subsets of P(U).

Step 4: Check the Identity Law

Verify the Identity law is true for the subsets of P(U). A ∪ ∅ = A and A ∩ U = A The Identity law is satisfied for all subsets of P(U).

Step 5: Check the Distributive Law

Verify the Distributive law is true for the subsets of P(U). A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) and A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) The Distributive law is satisfied for all subsets of P(U).

Step 6: Check the Complement Law

Verify the Complement law is true for the subsets of P(U). A ∪ A' = U and A ∩ A' = ∅ The Complement law is not satisfied for all subsets of P(U). Since the Complement Law is not satisfied for all subsets of P(U), the given structure 𝑆 is not a Boolean algebra.

Key Concepts

Power Set

The concept of a power set is foundational in set theory. When we talk about the power set of a given set, we are referring to the set that includes all possible subsets of the original set. It also includes the empty set and the set itself. For instance, if we have a set \(U = \{a, b, c\}\), its power set, denoted as \(P(U)\), would be:

• \(\emptyset\) (Empty Set)
• \(\{a\}\), \(\{b\}\), \(\{c\}\) (Single element subsets)
• \(\{a, b\}\), \(\{a, c\}\), \(\{b, c\}\) (Two elements subsets)
• \(\{a, b, c\}\) (The set itself)

Creating a power set is like playing a game of all possible combinations, where every single possibility is included. It's a complete universe of options within the boundaries of the original set.

Commutative Law

The commutative law is a principle found in mathematics that states the order in which you apply a binary operation does not change the result. In the context of set theory, this law applies to the operations of union (\(\cup\)) and intersection (\(\cap\)). For example:

• Union: \(A \cup B = B \cup A\), meaning adding set \(A\) to \(B\) is the same as adding \(B\) to \(A\).
• Intersection: \(A \cap B = B \cap A\), which means the common elements of \(A\) and \(B\) remain the same regardless of the order.

This law highlights the flexibility of combining sets, ensuring that the sequence doesn't matter. It helps simplify calculations and reasoning when working with sets.

Associative Law

The associative law, like the commutative law, is another important property in mathematics applied to operations such as union and intersection. This law implies that when we group three or more elements and perform an operation, the outcome remains unaffected by the grouping of those elements. For example:

• Union: \((A \cup B) \cup C = A \cup (B \cup C)\)
• Intersection: \((A \cap B) \cap C = A \cap (B \cap C)\)

In simple terms, this means the structure of how we group the sets doesn't change the resulting set when performing union or intersection. The associative law, therefore, aids in rearranging and simplifying set operations, making it easier to handle complex expressions.

Identity Law

Identity laws help define simple truths within mathematics, particularly in set operations. For sets, they explain how certain operations will leave a set unchanged when combined with specific elements. For example:

• Union: \(A \cup \emptyset = A\), meaning adding the empty set to any set \(A\) leaves \(A\) unchanged.
• Intersection: \(A \cap U = A\), where \(U\) is the universal set containing all possible elements within the context, intersecting any set \(A\) with \(U\) yields \(A\) itself.

These identity laws provide fundamental rules for interacting with sets, ensuring that certain set operations result in predictable and unchanged outcomes. They serve as a base for understanding more complex operations in set theory.