Problem 33 - Algebra for College Students

Question

List the elements of each set. For example, the elements of \(\\{x \mid x\) is a natural number less than 4\(\\}\) can be listed as \(\\{1,2,3\\}\). \(\\{x \mid x\) is a natural number less than 3\(\\}\)

Step-by-Step Solution

Answer

\(\{1, 2\}\)

Step 1: Understanding Natural Numbers

Natural numbers are counting numbers starting from 1, 2, 3, and so on. They do not include zero or any negative numbers.

Step 2: Interpreting the Set Condition

The condition in the set, 'natural number less than 3', means we need to find all natural numbers that are less than the number 3.

Step 3: Identifying the Elements

Since we are looking for natural numbers less than 3, we start counting from 1 upwards until we reach a number that is not less than 3. The numbers 1 and 2 satisfy this condition.

Step 4: Listing the Elements

The elements of the set are listed by writing down the numbers identified in the previous step. Hence, the elements of the set are \(\{1, 2\}\).

Key Concepts

Natural Numbers

Natural numbers are the numbers you use to count regular, everyday items. They start from 1 and continue indefinitely (i.e., 1, 2, 3, 4, etc.). It’s important to remember that natural numbers do not include zero, fractions, or any negative numbers. Think of them as the numbers you would naturally think of when counting objects, like apples or books.

• Natural numbers begin at 1.
• They are infinite; no largest natural number exists.
• They exclude zero and negative numbers.

Understanding natural numbers is crucial. They form the basis for more complex mathematical concepts. They are commonly used in everyday counting and in various mathematical operations and problems.

Elements of a Set

In set theory, elements are the distinct objects that belong to a particular set. A set is a collection of different elements. To determine the elements of a set, consider the rule or condition that defines the set. If a set is defined as consisting of natural numbers less than 3, then you evaluate which numbers satisfy this condition. Each number or object that fulfills the set's condition is an element of that set. For example, in the set defined as "natural numbers less than 3," the numbers 1 and 2 are elements.

• Elements must satisfy the set's defining conditions.
• Every element is unique in its set.
• The order in which elements are listed doesn't change the set.

Recognizing the elements of a set helps in understanding and solving various mathematical and logical problems.

Mathematical Sets

Mathematical sets are fundamental concepts in mathematics and are used to group together related objects. A set is a collection of elements, often defined by a particular property or rule. Elements in a set do not repeat and have no specific order. Sets are usually denoted using curly braces, like \( \{1, 2, 3\} \).

• Sets can be finite or infinite.
• They can contain numbers, objects, or even other sets.
• The description of the set determines its elements.

An example of a mathematical set in action is the exercise you’ve encountered: listing natural numbers less than 3. This concept highlights how sets can simplify and organize information, making it easier to work with groups of numbers or objects in mathematical problems.