Problem 67 - Discrete Mathematics with Applications

Question

State the distributive laws using the sets \(A\) and \(B_{i}, i \in I\)

Step-by-Step Solution

Answer

The two distributive laws for sets A and sets $B_i, i \in I$ are: 1. \(A \cap (\bigcup_{i \in I} B_i) = \bigcup_{i \in I} (A \cap B_i) \) - Intersection over Union 2. \(A \cup (\bigcap_{i \in I} B_i) = \bigcap_{i \in I} (A \cup B_i) \) - Union over Intersection

Step 1: Distributive Law 1: Intersection over Union

To state the distributive law for the intersection over the union of sets, we write the formula as follows: \[A \cap (\bigcup_{i \in I} B_i) = \bigcup_{i \in I} (A \cap B_i) \] This law states that the intersection of set A with a union of multiple sets Bi is equivalent to the union of the intersections of set A with each individual set Bi.

Step 2: Distributive Law 2: Union over Intersection

To state the distributive law for the union over the intersection of sets, we write the formula as follows: \[A \cup (\bigcap_{i \in I} B_i) = \bigcap_{i \in I} (A \cup B_i) \] This law states that the union of set A with an intersection of multiple sets Bi is equivalent to the intersection of the unions of set A with each individual set Bi.

Key Concepts

Distributive Laws

In set theory, distributive laws help us understand how operations of intersection and union interact with each other. These laws function much like distributive laws in arithmetic, where multiplication distributes over addition. In the context of sets:

• The first law, known as "Intersection over Union," is explained by the equation: \[ A \cap (\bigcup_{i \in I} B_i) = \bigcup_{i \in I} (A \cap B_i) \]. This means if you have a set \(A\) and you are intersecting it with a large union of many sets \(B_i\), it is the same as taking \(A\) and intersecting it individually with each \(B_i\), and then uniting all those results.
• The second law, "Union over Intersection," is expressed as: \[ A \cup (\bigcap_{i \in I} B_i) = \bigcap_{i \in I} (A \cup B_i) \]. Here, if set \(A\) is united with an intersection of several sets \(B_i\), we find it equivalent to uniting \(A\) with each \(B_i\) first, then obtaining the intersection of all these unions.

These distributive laws are crucial as they provide a way to simplify complex expressions involving unions and intersections, making problem-solving easier and more intuitive.

Intersection of Sets

The concept of intersection in set theory represents the common elements shared among sets. When you "intersect" sets, you are finding all items that the sets share.
For example, if you have Set A containing elements {1, 2, 3} and Set B containing {2, 3, 4}, their intersection, written as \(A \cap B\), would result in a new set {2, 3}, because those are the elements both sets have in common.

• Intersection is denoted by the symbol \(\cap\).
• It is commutative, meaning \(A \cap B = B \cap A\).
• If there are no common elements, the intersection is an empty set, denoted by \(\emptyset\).

Understanding intersection is essential for using the distributive laws effectively since these laws rely heavily on combining intersections with unions in meaningful ways.

Union of Sets

In contrast to intersection, the union of sets involves combining all the elements from different sets into one set without repeating any elements. It effectively gathers together everything from the sets involved.
For instance, if Set A contains {1, 2, 3} and Set B contains {2, 3, 4}, their union, noted as \(A \cup B\), becomes {1, 2, 3, 4} as it includes all unique elements from both sets.

• Union is represented using the symbol \(\cup\).
• The operation is associative \((A \cup B) \cup C = A \cup (B \cup C)\).
• The union enjoys a commutative property, which tells us \(A \cup B = B \cup A\).

Mastering the concept of unions is vital, especially when applying distributive laws, as these rules organize how unions and intersections can be seamlessly interchanged while preserving equalities and set identities.