Problem 7 - Calculus for Biology and Medicine

Question

Assume that $$ \Omega=[1,2,3,4,5,6\\} $$ \(A=\\{1,3,5\\}\), and \(B=\\{1,2,3\\}\) Find \((A \cup B)^{c}\).

Step-by-Step Solution

Answer

\((A \cup B)^{c} = \{4, 6\}\).

Step 1: Find A union B

The union of sets A and B, denoted by \(A \cup B\), includes all elements that are in either A, B, or both. Here, set \(A = \{1, 3, 5\}\) and set \(B = \{1, 2, 3\}\). Thus, the union \(A \cup B = \{1, 2, 3, 5\}\).

Step 2: Find the complement of A union B

The complement of a set \(A \cup B\), denoted by \((A \cup B)^{c}\), includes all elements that are not in \(A \cup B\) but are in the universal set \(\Omega\). The universal set \(\Omega = \{1, 2, 3, 4, 5, 6\}\) and \(A \cup B = \{1, 2, 3, 5\}\). Therefore, the complement of the union is \((A \cup B)^{c} = \{4, 6\}\).

Key Concepts

Union of Sets

The union of two sets, often represented as \(A \cup B\), is a fundamental concept in set theory. It refers to a set that contains all of the elements from both sets, without duplication. To visualize this, imagine combining all of the contents of two baskets into one large basket. In mathematical terms:

• For sets \(A = \{1, 3, 5\}\) and \(B = \{1, 2, 3\}\), the union \(A \cup B\) will include all distinct elements from both sets.
• This means every element appearing in \(A\), \(B\), or both, will be in the union.

Therefore, the union \(A \cup B = \{1, 2, 3, 5\}\). Notice how elements \(1\) and \(3\) appear in both sets but are listed only once in the union. The union operation is a way to gather all elements from the specified sets into a single collection.

Complement of a Set

The complement of a set is another essential aspect of set theory. It includes all elements not in the specified set, but within a defined universal set. If the universal set is thought of as the entire world of possible elements, the complement gives you everything outside your current collection.

• The notation \((A \cup B)^{c}\) represents the complement of the union of sets \(A\) and \(B\).
• Here, the universal set \(\Omega = \{1, 2, 3, 4, 5, 6\}\), and \(A \cup B = \{1, 2, 3, 5\}\).

The complement operation looks for elements absent from the union but present in the universal set. Consequently, \((A \cup B)^{c} = \{4, 6\}\). Identifying the complement involves excluding the elements in \(A \cup B\) from the total pool provided by the universal set.

Universal Set

The universal set, denoted as \(\Omega\) in this context, comprises all potential elements under consideration for a particular problem or scenario. Imagine it as the biggest frame encompassing all objects you're evaluating within a particular discussion of set theory.

• In our example, \(\Omega = \{1, 2, 3, 4, 5, 6\}\), which encapsulates every possible element we are dealing with.
• It plays a vital role when finding complements of sets, as it provides the total elements from which the non-membership (or absence) is considered.

Every other set discussed, like \(A\) and \(B\), and their operations, like union or intersection, will find context relative to the universal set. Understanding how your sets relate to the universal set is crucial for grasping set complements or any other set operation nuance.