Problem 3 - Discrete Mathematics with Applications

Question

Let \(A=\\{a, e, f, g, i\rangle, B=\\{b, d, e, g, h\\}, C=\\{d, e, f, h, i\\},\) and \(U=\\{a, b, \ldots, k\\}\) Find each set. $$ C \cap A^{\prime} $$

Step-by-Step Solution

Answer

The intersection of set C and the complement of set A, denoted as \(C \cap A'\), is the set \(\{d, h\}\).

Step 1: Find the complement of set A with respect to U

In order to find the complement of set A, we need to determine all the elements in set U that are not in set A. Set A contains elements {a, e, f, g, i} and set U contains elements {a, b, c, d, e, f, g, h, i, j, k}. So the complement of A, denoted as \(A'\), can be found by taking the elements in U but not in A: \[ A' = \{b, c, d, h, j, k\} \]

Step 2: Find the intersection of set C and complement of set A

Now that we have found the complement of set A, we can now find the intersection of set C and \(A'\). The intersection of two sets is the set of elements that are common to both sets. Set C is given as {d, e, f, h, i}, and from step 1, the complement of A is {b, c, d, h, j, k}. The common elements in set C and \(A'\) are: \[ C \cap A' = \{d, h\} \] Therefore, the intersection of set C and the complement of set A is the set \(\{d, h\}\).

Key Concepts

Set Complement

In set theory, the complement of a set is an important concept that allows us to understand what elements are present in the universal set but not in the given set. Imagine you have a universal set, often symbolized as \( U \), which contains all possible elements related to a particular discussion or problem.

• The complement of a set \( A \), denoted as \( A' \), consists of everything in \( U \) that is not in \( A \).
• To find \( A' \), we simply subtract all the elements of \( A \) from \( U \).

For example, if \( A \) has elements \( \{a, e, f, g, i\} \) and \( U \) has elements \( \{a, b, c, d, e, f, g, h, i, j, k\} \), then:\[ A' = \{b, c, d, h, j, k\} \]These are the elements in \( U \) not found in \( A \).The set complement is a powerful operation, as it helps us figure out what is missing in a set when viewed against the backdrop of the universal set.

Set Intersection

The intersection of sets is about finding commonalities. Specifically, when we talk about the intersection of two sets, we are interested in identifying the elements that both sets share.

• For two sets, say \( C \) and \( A' \), their intersection is denoted as \( C \cap A' \).
• This sets up a new set containing only the elements present in both \( C \) and \( A' \).

Consider set \( C = \{d, e, f, h, i\} \) and \( A' = \{b, c, d, h, j, k\} \). To find their intersection, identify common elements within both sets:\[ C \cap A' = \{d, h\} \]These are the elements that \( C \) and the complement of \( A \) have in common. Intersections are quite useful as they allow us to explore overlaps and shared characteristics within different groups.

Universal Set

In set theory, the universal set, denoted as \( U \), serves as the boundary for all discussions concerning elements, sets, and operations. It includes every possible element within a particular context or problem.

• The universal set encompasses all other sets being considered, so each element related to the topic at hand is contained within \( U \).
• It essentially acts as a reference from which we define complements and other set-related operations.

For example, if you're working with different groups of letters from \( \{a, b, c, ..., k\} \), this entire range would constitute your universal set. Any subset, such as \( A \) or \( B \), draws its elements from this universal set:\[ U = \{a, b, c, d, e, f, g, h, i, j, k\} \]The versatile role of the universal set is crucial, as it provides context for operations like finding complements, unions, and intersections. Without it, these operations wouldn't have a defined logical scope.