Problem 53 - Intermediate Algebra

Question

The set of odd integers between 10 and 18

Step-by-Step Solution

Answer

\( \{11, 13, 15, 17\} \)

Step 1: Identify the Range

First, determine the integers that lie between 10 and 18. These integers are 11, 12, 13, 14, 15, 16, and 17.

Step 2: Recognize Odd Numbers

Recall that odd numbers are integers that cannot be divided exactly by 2 (they leave a remainder of 1 when divided by 2). In the list from the previous step, identify the odd numbers: 11, 13, 15, and 17.

Step 3: Construct the Set of Odd Integers

List the odd integers identified in the previous step within curly brackets to signify a set. Therefore, the set of odd integers between 10 and 18 is \( \{11, 13, 15, 17\} \).

Key Concepts

Odd Integers

When we talk about odd integers, we refer to whole numbers that cannot be divided evenly by 2. Instead, when you divide an odd integer by 2, there will be a remainder of 1. This unique property of odd numbers helps to easily distinguish them from even integers. For example:

• If you take the integer 7 and divide it by 2, the result is 3 with a remainder of 1, so 7 is an odd number.
• Similarly, for the number 13, dividing it by 2 gives 6 with a remainder of 1, confirming it's odd as well.

Recognizing odd numbers is about looking at the simplicity in division and understanding this fundamental characteristic.
Odd numbers have various interesting properties such as:

• The sum of two odd numbers is always even.
• When you multiply an odd number by another odd number, the result remains odd.
• The sequence of odd numbers starts from 1 and includes numbers like 3, 5, 7, and continues indefinitely.

Number Sets

In mathematics, a set is a collection of distinct objects, considered as an object in its own right. Using sets makes organizing and classifying numbers easier. For example, we often deal with different number sets such as the set of whole numbers, the set of integers, and more specifically, the set of odd integers as discussed earlier.
To create a set, we list its elements inside curly braces, e.g., \( \{a, b, c\} \). When dealing with number sets:

• Elements are unique, meaning a number cannot be repeated in a set.
• The order of elements in a set does not matter; \( \{3, 5\} \) is the same as \( \{5, 3\} \).
• We can specify a set by clearly defining its properties, like the set of integers between two given numbers.

Sets can be finite or infinite and help in simplifying complex mathematical concepts into more manageable parts.

Integer Properties

Integers are numbers without fractions or decimals, they include both positive and negative numbers as well as zero. Understanding integer properties is crucial in mathematics as they form the backbone of several mathematical operations and theories.
Key properties of integers include:

• Closure: Adding or multiplying two integers will always yield another integer. This is known as closure under addition and multiplication.
• Associativity: Grouping of integers doesn’t change their addition or multiplication result; for example, \( (a + b) + c = a + (b + c) \).
• Distributive Property: Multiplication distributes over addition; i.e., \( a(b + c) = ab + ac \).

These properties make integers a reliable and consistent set of numbers for performing operations. They enable the solving of equations and simplification of expressions in a logical way. Integers are pivotal in forming the rules for arithmetic operations and are foundational for understanding advanced mathematical principles.