Problem 1 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Write the set in set-builder notation. The set of gold medalists in the 2010 Winter Olympic Games

Step-by-Step Solution

Answer

\(G = \{ A : \text{A won a gold medal in the 2010 Winter Olympic Games} \}\)

Step 1: Define the set

We will define a set G which will represent the set of gold medalists in the 2010 Winter Olympic Games.

Step 2: Write the set-builder notation

Using the given information and our definition of A and property, we can represent the set G of gold medalists in the 2010 Winter Olympic Games using set-builder notation as: \(G = \{ A : \text{A won a gold medal in the 2010 Winter Olympic Games} \}\) This notation means "G is the set of all elements A, such that A won a gold medal in the 2010 Winter Olympic Games".

Key Concepts

Sets in Mathematics

Sets are a fundamental concept in mathematics, representing collections of distinct objects or elements. These objects, often numbers or symbols, can be grouped together based on common properties. Sets help organize information and are crucial in understanding various mathematical concepts.
To describe sets, we use curly brackets \{ \} and separate elements with commas. For example, a set of integers 1, 2, and 3 is denoted as \{1, 2, 3\}.
Sets can be finite or infinite. Finite sets contain a limited number of elements, while infinite sets can continue indefinitely, like the set of all whole numbers. Understanding sets also involves different types of sets, such as:

• Universal Set: Contains all possible elements relevant to a particular discussion.
• Subset: Every element of subset A is in set B, written as A \(\subseteq\) B.
• Intersection and Union: The intersection (\(A \cap B\)) includes elements common to both sets, while the union (\(A \cup B\)) consists of elements in either or both sets.

Sets form the backbone of many mathematical theories, allowing us to better understand relations and functions.

2010 Winter Olympic Games

The 2010 Winter Olympic Games were an international sporting event held in Vancouver, Canada. From February 12 to February 28, athletes from various countries competed in a range of winter sports. Events included skiing, ice hockey, figure skating, and more.
These games were memorable, showcasing outstanding athletic performances and celebrating sportsmanship and international cooperation. The 2010 Winter Olympics also played an essential role in raising awareness about winter sports and generating enthusiasm across the globe.

• Notable Achievements: The USA, Canada, and Germany led the medal tally, with Canada setting a record for the most gold medals in a single Winter Olympics by winning 14.
• Legacy: The games brought positive changes to Vancouver, including infrastructure improvements and ongoing economic benefits.

Understanding the historical context of the 2010 Winter Olympics can enrich our comprehension of the achievements and significance of the medalists involved.

Mathematical Notation

Mathematical notation refers to a system of symbols used to represent ideas, concepts, and relationships in mathematics. It provides a universal language for mathematicians and scientists, facilitating communication and understanding.
Notation varies across mathematical disciplines, but some general components include:

• Symbols: Common symbols include \(\forall\) (for all), \(\exists\) (there exists), and \(\in\) (element of).
• Numbers and Operators: Standard numerals, along with operators like + (plus), - (minus), and = (equals), form the building blocks of mathematics.
• Functions and Expressions: Functions denoted by symbols such as \(f(x)\) describe relationships between variables and are crucial in calculus and algebra.
• Set-Builder Notation: Uses a rule to define a set's elements, such as \(\{ x : x > 0 \}\), meaning the set of all x such that x is greater than 0.

Mastering mathematical notation helps break down complex problems into manageable parts, simplifying the process of finding solutions and communicating them effectively.