Problem 34 - Discrete Mathematics with Applications

Question

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

Step-by-Step Solution

Answer

The given set system $\langle S, \cup, \cap, \emptyset, U \rangle$ is indeed a Boolean algebra for the given subset $S$ of $P(U)$, as it satisfies all the axioms of a Boolean algebra.

Step 1: Identify the involved sets

We are given the following subset \(S\) of \(P(U)\): $$S = \{ \emptyset, \{c\}, \{a, b\}, U\}.$$

Step 2: Verify closure under union and intersection

To do this, we will find all possible unions and intersections of the sets in the given set \(S\): - \(O \cup O = O\) - \(O \cup \{c\} = \{c\}\) - \(O \cup \{a, b\} = \{a, b\}\) - \(O \cup U = U\) - \(\{c\} \cup \{a, b\} = \{a, b, c\} = U\) - \(\{c\} \cup U = U\) - \(\{a, b\} \cup U = U\) Notice that this ensures closure under the union operation. Now consider intersections: - \(O \cap O = O\) - \(O \cap \{c\} = O\) - \(O \cap \{a, b\} = O\) - \(O \cap U = O\) - \(\{c\} \cap \{a, b\} = O\) - \(\{c\} \cap U = \{c\}\) - \(\{a, b\} \cap U = \{a, b\}\) It is clear that this also ensures closure under the intersection operation.

Step 3: Check other axioms

To check the other axioms, we don't need to compute all the combinations again. Paying attention to axioms (3)-(8), it is obvious that all those axioms are naturally inherited from set theory and thus will be satisfied for any such set-based Boolean algebra. One thing to pay attention to is that there must be an element in \(S\) that acts as a negation for every element in \(S\). Let's verify this: - For \(\emptyset\), \(U\) is its negation. - For \(U\), \(\emptyset\) is its negation. - For \(\{c\}\), \(\{a,b\}\) is its negation. - For \(\{a,b\}\), \(\{c\}\) is its negation. All the axioms are satisfied, so:

Step 4: Conclusion

The given set system \(\langle S, \cup, \cap, \emptyset, U \rangle\) is indeed a Boolean algebra for the given subset \(S\) of \(P(U)\).

Key Concepts

Subset

A subset is a fundamental concept in set theory. It refers to a set whose elements are all contained within another set, known as the superset. In the context of the given exercise, we are looking at the subset \( S \) which is a subset of the power set \( P(U) \), where \( U = \{a, b, c\} \). This subset \( S \) consists of sets such as \( \emptyset, \{c\}, \{a, b\}, \) and \( U \) itself.

To put it simply, think of a subset as a group within a larger group. All members of the subgroup are members of the larger group. For example, \( \{c\} \) is a subset of \( U \), because \( c \) is an element of \( U \) which is \( \{a, b, c\} \).

In mathematical notation, we write \( A \subseteq B \) to indicate that \( A \) is a subset of \( B \). Here, \( A \) could be any set like \( \{c\} \), while \( B \) can be \( U \). This means every element of \( A \) is also an element of \( B \). Furthermore, every set is a subset of itself, and the empty set \( \emptyset \) is a subset of every set.

• All the elements in \( A \) are also in \( B \).
• An empty set is a subset of any set.
• A set is a subset of itself.

Set Theory

Set theory is a branch of mathematical logic that deals with sets, which are collections of objects. Sets are fundamental objects in mathematics, and set theory provides a unifying framework for understanding a wide range of mathematical concepts.

The basic idea is simple: if you can collect things together into a "set," then you can work with them as a single entity. For example, \( U = \{a, b, c\} \) can be seen as a single object that contains the items \( a \), \( b \), and \( c \).

In our exercise, we are given \( S \), a subset of \( P(U) \). The power set \( P(U) \) of a set \( U \) represents all possible subsets that can be formed from the elements of \( U \). Thus, the sets in \( S \) such as \( \{c\} \) and \( \{a, b\} \) are among those possible combinations.

• A set is a well-defined collection of distinct objects.
• Every element is either in a set or not; there is no middle ground.
• The power set of \( U \), \( P(U) \), includes every possible subset derived from \( U \).

Set theory also introduces operations such as union and intersection, which we'll explore next.

Union and Intersection

Union and intersection are two fundamental operations in set theory. These operations are crucial in Boolean algebra, as they help us combine and compare different sets.

The **union** of two sets combines all elements from both sets, without duplicating any elements. In mathematical terms, the union of sets \( A \) and \( B \) is denoted as \( A \cup B \). It includes every element from both \( A \) and \( B \). For example, if \( A = \{c\} \) and \( B = \{a, b\} \), then \( A \cup B = \{a, b, c\} \).

• The union accumulates elements from both sets.
• Duplicate elements are only represented once.

The **intersection** of two sets, on the other hand, includes only the elements common to both sets. It is denoted by \( A \cap B \). For example, with \( A = \{c\} \) and \( B = \{a, b\} \), \( A \cap B = \emptyset \) since there are no common elements. However, for \( B = \{a, b\} \) and \( U = \{a, b, c\} \), \( B \cap U = \{a, b\} \) because those are the elements both sets share.

• The intersection only includes elements present in both sets.
• If no common elements exist, the intersection is the empty set \( \emptyset \).