Problem 6 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

List the elements of the set in roster notation. $$ \\{x \mid x \text { is a letter in the word HIPPOPOTAMUS }\\} $$

Step-by-Step Solution

Answer

\( \{x \mid x \text { is a letter in the word HIPPOPOTAMUS }\} = \{H, I, P, O, T, A, M, U, S\} \)

Step 1: Identify distinct letters of the word HIPPOPOTAMUS

To list the elements in the set, identify all the distinct letters in the given word. The word HIPPOPOTAMUS contains the following distinct letters: H, I, P, O, T, A, M, U, S.

Step 2: Organize letters using roster notation

Now that we have identified all the distinct letters in the word HIPPOPOTAMUS, the next step is to list these letters using roster notation within curly brackets {}. The required set is: \[ \{H, I, P, O, T, A, M, U, S\} \] So the final answer is: \[ \{x \mid x \text { is a letter in the word HIPPOPOTAMUS }\} = \{H, I, P, O, T, A, M, U, S\} \]

Key Concepts

Set Theory

Set theory is a fundamental part of mathematics which deals with the concept of a collection of objects, known as sets. A set can be defined as a grouping of distinct objects considered as a whole. These objects are called the 'elements' or 'members' of the set.

The significance of set theory lies in its role as the foundation for various other fields of mathematics, including algebra, geometry, and probability. It provides a unifying framework for understanding different mathematical concepts by analyzing how they can be collected into sets and how these sets relate to one another.

In our example with the word HIPPOPOTAMUS, we are looking at a finite set, meaning it has a limited number of elements. To define this set in a comprehensible way, we identify all the unique letters in the word to form a set that contains these letters as its elements. Set theory facilitates the classification and organization of these elements, allowing mathematicians to work with them systematically.

Distinct Elements

In mathematics, particularly in set theory, distinct elements refer to individual items in a set that are unique and non-repeating. In the context of our exercise, the distinct elements are the individual letters of the word 'HIPPOPOTAMUS' that do not duplicate within the set.

Identifying distinct elements is crucial because in a set, repetition of elements is irrelevant; what matters is whether an item is a member of the set. In simple terms, in set theory, a set containing the letters {A, A, B} is considered the same as a set containing {A, B} — the repetition of the letter 'A' doesn’t impact the identity of the set.

Thus, when listing the elements of the set derived from HIPPOPOTAMUS, we ensure that each letter appears only once, regardless of how many times it might appear in the original word. This approach highlights the importance of distinguishing each element in a set for clarity and accuracy in mathematical notation and operations.

Mathematical Notation

Mathematical notation is the system of symbols and notations used to express mathematical ideas, operations, and relations. It is the language through which mathematicians communicate concepts in a precise and unambiguous manner.

Roster notation is a method of representing a set by listing its elements between curly brackets. For example, \( \{H, I, P, O, T, A, M, U, S\} \) is the roster notation for the set of distinct letters in the word 'HIPPOPOTAMUS'. Using such notation facilitates the understanding and analysis of sets, making it clear to anyone studying the problem what the set contains.

In addition to roster notation, other common forms of mathematical notation include set-builder notation, which defines a set by a property that its members must satisfy, and interval notation, which is often used for denoting ranges of numbers. Mastering these notations is essential for students as these are the tools they will use to bridge the gap between abstract mathematical ideas and their concrete representations.