Problem 5 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

List the elements of the set in roster notation. $$ \\{x \mid x \text { is a digit in the number } 352,646\\} $$

Step-by-Step Solution

Answer

The unique digits in the number 352,646 are: 3, 5, 2, 6, and 4. In roster notation, the elements of the given set can be written as: \(\{3, 5, 2, 6, 4\}\).

Step 1: Identify the unique digits

To list the elements of the set, we first need to identify the unique digits present in the given number, 352,646. To do this, simply look at the digits and identify if they are distinct.

Step 2: List the unique digits

The unique digits in the number 352,646 are: 3, 5, 2, 6, and 4. (Even though the digit 6 appears twice, it is considered as a single unique digit for the set.)

Step 3: Write in roster notation

In roster notation, we list the elements of a set within curly brackets, with each element separated by a comma. Thus, the elements of the given set can be written as: \[ \{3, 5, 2, 6, 4\} \] Thus, the elements in the set in roster notation are: \(\{3, 5, 2, 6, 4\}\).

Key Concepts

Set Theory

Set theory is a fundamental part of mathematics that deals with the collection of objects, known as elements, and the relationships between these objects. In the context of set theory, a set is defined as a well-defined collection of distinct objects, and these objects can be anything – numbers, letters, symbols, or even other sets.

Considering our exercise, we needed to identify the unique elements, or digits, from a number sequence to form a set. This is much like how a fruit bowl might contain only one type of each fruit. Furthermore, in set theory, repetition is not recognized; each element is unique. So, even if a digit appears multiple times within the number sequence, it is only counted once to represent a set of unique digits.

Unique Digits

Speaking of unique digits, the concept of uniqueness is a key component in set theory. Each element or digit in a set must be distinct. As with the number sequence given in our exercise, 352,646, we find that while some digits repeat, we consider each only once for the set. This rule helps avoid redundancy and ensures that sets are precise and concise.

Just like every person has a unique fingerprint, each digit in our set must be different from the others. When we identify 'unique digits,' we're similar to detectives, sifting through the sequence, and choosing only one representative of each number. In our exercise, '6' is a repeat offender but gets listed only once in our collection of suspects.

Mathematical Notation

Finally, mathematical notation is like the language of mathematics; it's a system of symbols that communicate mathematical ideas succinctly and accurately. In terms of sets, roster notation is one way to express a set, where we list every element in curly brackets and separate them with commas.

For our exercise featuring the set \( \{3, 5, 2, 6, 4\} \), the roster notation clearly communicates each member of the set without ambiguity. This practice eliminates confusion and allows mathematicians and students alike to understand exactly what the set includes. Just as grammar rules are crucial for crafting understandable sentences, roster notation follows specific guidelines that ensure sets are presented clearly.