Question

(1) Find set \(\mathrm{S}=\mathrm{A} \cup(\mathrm{B} \cap \mathrm{C})\) where \(\mathrm{U}=\\{2,4,6,8,10, \mathrm{x}, \mathrm{y}, \mathrm{z}\\}, \quad \mathrm{A}=\\{2,4, \mathrm{x}, \mathrm{y}\\}\) \(\mathrm{B}=\\{2,4,6,8,10\\}, \quad\) and \(\quad \mathrm{C}=\\{6,8, \mathrm{z}\\}\) (2) Draw the Venn Diagram of the set \(A \cup(B \cap C)\).

Short answer

The set $S=A \cup(B \cap C)$ can be calculated by using the given sets $A = \{2, 4, x, y\}$, $B = \{2, 4, 6, 8, 10\}$, and $C = \{6, 8, z\}$. First, find the intersection of $B$ and $C$ which is $B \cap C = \{6, 8\}$. Now, the union of set $A$ and the intersection set $(B \cap C)$ is $S = A \cup (B \cap C) = \{2, 4, 6, 8, x, y\}$. The Venn diagram for $A \cup (B \cap C)$ consists of three overlapping circles for sets $A$, $B$, and $C$ with the universal set $U = \{2, 4, 6, 8, 10, x, y, z\}$.

Step-by-step solution

Find the intersection of sets B and C

To find the intersection of sets B and C, we need to find the common elements between B and C. The intersection set can be denoted as B ∩ C. B = {2, 4, 6, 8, 10} C = {6, 8, z} B ∩ C = {6, 8}

Find the union of set A and the intersection set (B ∩ C)

Now that we have found the intersection of sets B and C, we need to find the union of set A and the intersection set (B ∩ C). A = {2, 4, x, y} (B ∩ C) = {6, 8} The union set can be denoted as A ∪ (B ∩ C) or simply S. To find the union set, we combine all the elements of set A and the intersection set (B ∩ C). A ∪ (B ∩ C) = {2, 4, 6, 8, x, y} S = {2, 4, 6, 8, x, y}

Draw the Venn diagram of the set A ∪ (B ∩ C)

To draw the Venn diagram of the set A ∪ (B ∩ C), follow these steps: 1. Draw three overlapping circles representing sets A, B, and C. 2. Label the circles with their respective set names. 3. Fill the intersection of set B and C with the elements {6, 8}. 4. Fill the remaining spaces in set A with its unique elements {2, 4, x, y}. 5. Enclose the sets with a rectangle representing the universal set U. The Venn diagram should look like this: U: {2, 4, 6, 8, 10, x, y, z} A: {2, 4, x, y} B: {2, 4, 6, 8, 10} C: {6, 8, z} The desired set A ∪ (B ∩ C) is represented by the union of A and the intersection of B and C, which contains the elements {2, 4, 6, 8, x, y}.

Key concepts

Venn Diagrams

Venn diagrams are a visual way to show the relationships between different sets. They use circles to represent each set and place them in a way that highlights the commonalities and differences between the sets. Each circle in a Venn diagram represents one of the sets, and overlapping areas show the elements that two or more sets have in common. For example, if we have three sets, A, B, and C, we draw three overlapping circles. The regions where the circles overlap represent the intersections of these sets.

Venn diagrams are particularly useful because they provide a clear and intuitive way to visualize set operations. They help us quickly see how elements are shared among sets, making it easier to understand concepts like unions and intersections. In the exercise, by drawing a Venn diagram of sets A, B, and C, we can clearly identify the elements that belong to set S, which is the union of A and the intersection of B and C.

Set Operations

Set operations are mathematical actions that we can perform on two or more sets to get a new set. The most common set operations are union, intersection, and difference. These operations allow us to combine sets in various ways. Each operation has its own specific result depending on the elements contained within the original sets. For instance, if we were to perform the union operation on sets A and B, we'd include all elements from both set A and set B without repeating any elements.

In set theory, these operations are fundamental because they form the basis for defining and working with different types of mathematical objects. By understanding set operations, we can solve problems that involve combining, separating, and altering sets, making it easier to analyze relationships and connections between different groups of elements.

Intersection of Sets

The intersection of sets is a key operation in set theory that results in a new set containing only the elements that are present in both original sets. We use the symbol \( \cap \) to denote intersection. For example, if set B contains elements \( \{2, 4, 6, 8, 10\} \) and set C contains \( \{6, 8, z\} \), the intersection \( B \cap C \) would be the set \( \{6, 8\} \) since both sets have these elements in common.

This concept is important as it helps to pinpoint the exact elements that belong to both sets, effectively illustrating their commonalities. In the problem, finding \( B \cap C \) was a crucial step in constructing set S, demonstrating the importance of understanding intersections.

Union of Sets

The union of sets is another fundamental operation in set theory. It combines all the elements from the given sets into a single new set. The union is denoted by the symbol \( \cup \). When we calculate a union, we include every distinct element that appears in any of the participant sets.

For example, to find the union of set A, which contains \( \{2, 4, x, y\} \), and the intersection set \( B \cap C \), which contains \( \{6, 8\} \), we form the set \( A \cup (B \cap C) \) as \( \{2, 4, 6, 8, x, y\} \). This operation helps to quickly gather all unique elements from multiple sets, offering a comprehensive view of what items each set contributes to the union.