Problem 1 - 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. $$ C^\prime $$

Step-by-Step Solution

Answer

The complement of set C, denoted as \(C'\), is the set of elements in the universal set U that are not in C. Given that \(U = \{a, b, c, d, e, f, g, h, i, j, k\}\) and \(C = \{d, e, f, h, i\}\), we can find \(C'\) by identifying the elements in U that are not in C. Therefore, \(C' = \{a, b, c, g, j, k\}\).

Step 1: Understand the universal set U

The universal set U is given as {a, b, ..., k}, which means it contains every letter from a to k. So, U = {a, b, c, d, e, f, g, h, i, j, k}. #Step 2: Identify the elements in the set C#

Step 2: Understand the set C

Set C is given as {d, e, f, h, i}. #Step 3: Find the complement of set C#

Step 3: Calculate the complement of C

To find the complement of C, look for elements in U that are not in C. The elements 'a', 'b', 'c', 'g', 'j', and 'k' are not present in the set C. So, the complement of C, written as \(C'\), is the set of these elements. Therefore, \(C' = \{a, b, c, g, j, k\}\).

Key Concepts

Universal Set

The universal set, often denoted by the symbol \( U \), is a fundamental concept in set theory. It encompasses all possible elements under consideration within a particular context or problem. In simpler terms, the universal set contains every item we may discuss in a particular situation. For instance, if our universe is focused on the English alphabet, then \( U \) would consist of the letters from 'a' to 'z'.

In the given exercise, the universal set is specified as \( U = \{a, b, c, d, e, f, g, h, i, j, k\} \). This means our universe includes all these elements, and no element beyond this list is considered for this particular problem. Understanding which elements are included in \( U \) is crucial when exploring other set-related operations, such as finding complements, unions, or intersections.

A simple way to think about the universal set is to consider it as the 'everything' of our discussion: it includes every possible element we are interested in talking about. This concept helps set the stage for determining what is included or excluded in specific smaller subsets like \( A \), \( B \), or \( C \). This understanding paves the way to successfully perform set operations.

Complement of a Set

The complement of a set is like the opposite of that set, relative to a universal set. In other words, it consists of elements that are in the universal set \( U \) but not in the original set we are examining. This is symbolized by a prime (\( ' \)) symbol next to the set's name, such as \( C' \) for the complement of set \( C \).

To find the complement, we identify all the elements in the universal set \( U \) that are not present in our particular set. In our exercise, the task is to find \( C' \) where \( C = \{d, e, f, h, i\} \). Knowing \( U = \{a, b, c, d, e, f, g, h, i, j, k\} \), find the elements missing from \( C \). These elements are 'a', 'b', 'c', 'g', 'j', and 'k'. Thus, \( C' \) equals those elements, written as \( C' = \{a, b, c, g, j, k\} \).

Understanding the concept of complement is crucial because it helps to see what is outside a particular set, offering a broader view of a universal set minus a subset's perspective.

Set Notation

Set notation is the language of sets, and it helps us describe the members of a set clearly and precisely. When using set notation, we typically write sets with curly braces \( \{ \} \), listing their elements separated by commas.

This convention allows us to quickly and efficiently convey the elements that are included within a set. For instance, in our example problem, each of the sets is expressed in clear set notation:

• Set \( A \) is \( \{a, e, f, g, i\} \)
• Set \( B \) is \( \{b, d, e, g, h\} \)
• Set \( C \) is \( \{d, e, f, h, i\} \)
• The universal set \( U \) is \( \{a, b, c, d, e, f, g, h, i, j, k\} \)

Set notation allows for precise identification and manipulation of sets, facilitating operations like unions, intersections, and compliments. Being comfortable with set notation is essential for anyone looking to tackle problems in set theory efficiently.