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Chapter 5: Problem 8

Determine the cardinality of the given set..$$4 x+3=31$$ \(n(H)\) where \(H=\\{x | x \text { is an odd number and } 1

Short Answer

Expert verified
The cardinality of set \(H\) is 6.

Step by step solution

01

Solve the Equation

To find the value of \(x\), solve the equation: \[4x + 3 = 31\]Subtract 3 from both sides:\[4x = 28\]Now, divide both sides by 4:\[x = 7\]
02

Define Set H

Set \(H\) is defined as the set of all odd numbers where \(1 < x \leq 14\). So we need to list odd numbers that lie between 1 and 14, inclusive of 14.
03

List Odd Numbers in Range

List all odd numbers between 1 and 14: \[3, 5, 7, 9, 11, 13\]
04

Determine Cardinality of Set H

The cardinality of a set is the number of elements in the set. Count the elements in \(H\), which are \{3, 5, 7, 9, 11, 13\}. There are 6 numbers, so the cardinality is 6.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Solving Linear Equations
Linear equations are equations of the first degree, which means they have variables raised only to the power of one. The standard form is generally expressed as \(ax + b = c\). Here, we are given the equation \(4x + 3 = 31\). To solve this, we need to isolate the variable \(x\).
  • First, subtract 3 from both sides of the equation: \(4x + 3 - 3 = 31 - 3\), which simplifies to \(4x = 28\).
  • Next, divide both sides by 4 to solve for \(x\): \(4x/4 = 28/4\), giving us \(x = 7\).
Finding the solution is crucial as it helps identify specific values that satisfy the equation.
Defining Sets
A set is a collection of distinct objects, considered as an object in its own right. Sets are usually defined by listing their elements or by a property that their elements must satisfy. The set in the problem, \(H\), is defined by a property: it includes all odd numbers where \(1 < x 鈮� 14\).
  • We specify sets using curly braces and can express them both by listing elements or by property notation, like \(\{x | 1 < x 鈮� 14\}\).
  • In this exercise, the elements of set \(H\) need to satisfy the condition of being odd numbers within a specified range.
Defining sets clearly is important to avoid confusion and to work with precise mathematical objects.
Odd Numbers
Odd numbers are integers not divisible by 2. When defining a set of odd numbers within a given range, we focus on those numbers that fit the criteria set by the range. In our problem, the set includes odd numbers greater than 1 and up to 14.
  • The sequence of odd numbers is: 1, 3, 5, 7, etc.
  • For the set \(H\), we specifically look at numbers \(3, 5, 7, 9, 11, 13\).
  • These numbers maintain the requirement of being greater than 1 and not exceeding 14.
Understanding odd numbers is crucial as they form an essential part of number theory and help in solving problems with specific number types.
Range of Numbers
A range in mathematics refers to a set of numbers that satisfy certain conditions, usually defined by lower and upper bounds. Here, the range is \(1 < x \leq 14\). This means we look for numbers greater than 1 and up to 14.
  • The range itself is an interval that helps define the boundaries within which numbers can be included in a set.
  • The condition affects which numbers meet the criteria to belong to the set \(H\).
Understanding the concept of a range helps you determine which numbers are inside or outside of a set, guiding you in identifying set membership.

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