Problem 5 - Applied Discrete Structures

Question

What can you say about \(A\) if \(U=\\{1,2,3,4,5\\}, B=\\{2,3\\},\) and \((\) separately) (a) \(A \cup B=\\{1,2,3,4\\}\) (b) \(A \cap B=\\{2\\}\) (c) \(A \oplus B=\\{3,4,5\\}\)

Step-by-Step Solution

Answer

The set \(A\) is \{2,4,5\}. It satisfies all conditions given in the problem.

Step 1: Understanding Universal Set and Sets B

The universal set given is \(U=\{1,2,3,4,5\}\) and another set \(B=\{2,3\}\). We need to analyze different scenarios for set \(A\) based on the operations provided.

Step 1: Analyze Union Operation

Given \(A \cup B = \{1,2,3,4\}\), this means the combination of elements from both sets \(A\) and \(B\) results in \{1,2,3,4\}. Since \(B\) contains \{2,3\}, \(A\) must contain at least \{1,4\}. Therefore, \(A\) could be \{1,4\}, or \{1,4,5\}.

Step 2: Analyze Intersection Operation

Given \(A \cap B = \{2\}\), it suggests that the element \(2\) is the only common element between \(A\) and \(B\). Given \(B=\{2,3\}\), \(A\) must at least contain \{2\} but must not contain \{3\}.

Step 3: Analyze Symmetric Difference Operation

Given \(A \oplus B = \{3,4,5\}\), this operation implies elements that are in \(A\) or \(B\) but not both. Since \(B=\{2,3\}\), then \(A\) must include \{4,5\} (elements from \{3,4,5\} not in \(B\)) and exclude \{2\}, because \{2\} is common.

Step 5: Conclusion

Based on the operations, the set \(A\) must be \{2,4,5\} to satisfy all given conditions: it provides elements \{4\} and \{5\} for the symmetric difference, element \{2\} for the intersection, and combines with \(B\) to yield \{1,2,3,4\} for the union (since 1 must come from \(A\)).

Key Concepts

Universal Set

In set theory, a universal set, often denoted as \( U \), is the set that contains all the objects or elements under consideration for a particular discussion or problem. For example, if you are determining subsets within a collection of numbers, your universal set could encompass all the numbers within a specified range. In this exercise, the universal set is given as \( U = \{1, 2, 3, 4, 5\} \). This set embodies the entire scope of numbers we are working with, meaning any other set mentioned in the problem (like set \( A \) or set \( B \)) will exclusively include elements drawn from \( U \). Understanding the universal set is crucial because it acts as the framework within which all our other set operations take place. It is the 'big container' that holds every element we consider in the problem.

Union Operation

The union operation, denoted by \( \cup \), combines all the elements from two sets, including all their individual members without repetition. If you imagine two circles representing two sets, the union represents everything within both circles. In mathematical terms, for sets \( A \) and \( B \), the union \( A \cup B \) includes every element in \( A \), as well as every element in \( B \).

• For example, if \( A = \{1, 4\} \) and \( B = \{2, 3\} \), their union would be \( A \cup B = \{ 1, 2, 3, 4 \} \).

In this exercise, we're told \( A \cup B = \{1, 2, 3, 4\} \). Since set \( B \) contains \{2, 3\}, the remaining elements \{1, 4\} must come from set \( A \). Hence, \( A \) could be \( \{1, 4\} \), or it could include additional elements like \{5\}, but must include at least \{1, 4\} to satisfy the union result.

Intersection Operation

The intersection operation, symbolized by \( \cap \), identifies common elements that are shared between two sets. Imagine two overlapping circles representing the sets; the intersection is the area where the circles overlap. This operation is foundational in set theory as it focuses on similarities between sets rather than differences. Mathematically, for sets \( A \) and \( B \), the intersection \( A \cap B \) is a set of all elements \( x \) such that \( x \in A \) and \( x \in B \).

• In this exercise, \( A \cap B = \{2\} \) means that \( 2 \) is the only element that both sets share.

Given that set \( B = \{2, 3\} \), set \( A \) must include 2, but must not carry 3 as part of it to maintain the intersection as only \{2\}. The intersection informs us about the strict commonality between \( A \) and \( B \), thereby guiding us on which elements cannot coexist in both sets simultaneously if not desired.

Symmetric Difference

The symmetric difference, represented by \( \oplus \), relates to elements that are part of either set but not part of their intersection. It's akin to finding differences rather than similarities in set theory. Think of it as highlighting only the non-overlapping parts of two sets. For sets \( A \) and \( B \), the symmetric difference \( A \oplus B \) includes all elements that are in \( A \) or in \( B \), but not in both.

• Consider \( B = \{2, 3\} \) from the exercise, and we know \( A \oplus B = \{3, 4, 5\} \). This suggests that \( A \) must include \{4, 5\} as unique elements from the symmetric difference, while avoid including \{2\} since it forms the intersection.

Ensuring that only non-common elements are present in the symmetric difference is crucial to solving problems like this, where both sets' individualized content is in focus rather than shared elements.