Problem 29 - College Algebra with Corequisite Support

Question

For the following exercises, find the number of subsets in each given set. The set of two-digit numbers between 1 and 100 containing the digit 0

Step-by-Step Solution

Answer

The set has 512 subsets.

Step 1: Identify Elements Containing '0'

To find the subsets of the set containing two-digit numbers with the digit '0', first list all two-digit numbers between 1 and 100 that contain the digit '0'. These numbers are: 10, 20, 30, 40, 50, 60, 70, 80, 90.

Step 2: Count the Elements in the Set

Now that we have identified the elements, count them. In this case, there are 9 numbers: 10, 20, 30, 40, 50, 60, 70, 80, and 90.

Step 3: Calculate the Number of Subsets

The number of subsets of a set is found using the formula \( 2^n \), where \( n \) is the number of elements in the set. Here, \( n = 9 \), so the number of subsets is \( 2^9 = 512 \).

Key Concepts

Subsets

A subset is a collection of elements that are all contained within another set. In simpler terms, if you have a set containing a variety of items, any grouping of those items can be considered a subset. Some important things to know about subsets include:

• Every set is itself a subset, which is sometimes called the "improper subset."
• An empty set, which contains no elements, is also a subset of every set.
• The number of possible subsets of a set depends on the number of elements in it.

To figure out the number of possible subsets, we can use a power of 2 formula. If a set has \( n \) elements, then it has \( 2^n \) subsets. This formula takes into account all the different ways you can combine each element with every other element or leave them out entirely.

Two-digit numbers

Two-digit numbers are numbers ranging from 10 to 99. These numbers are significant as they form the building blocks of counting and are easily recognizable by their two-place digits. In this context of set theory, we focus on two-digit numbers that contain specific digits—in this problem, the digit '0'. Some things to keep in mind about two-digit numbers involving the digit 0:

• They must have two places or digits, where the digit '0' can appear in the tens or units place.
• In the range from 10 to 99, only the tens place can contain a '0', giving potential numbers like 10, 20, up to 90.

These numbers not only help us in practicing our basic mathematics but also provide a good foundation when learning new concepts, such as understanding how specific digits appear in arrays of numbers.

Counting elements

Counting elements refers to determining how many items are present in a particular set. This is a basic but crucial skill in set theory and math in general. To successfully count elements in a specific set, follow these steps:

• Identify which items or numbers are part of the set. In our case, these are the two-digit numbers containing the digit 0: 10, 20, 30, 40, 50, 60, 70, 80, and 90.
• Count each distinct item or number once. The order doesn't matter; what we care about is the total count.

For our set, this results in a total count of 9 elements. Counting is straightforward but pivotal when you move on to more complex set manipulations like finding subsets.

Mathematical formulas

Mathematical formulas provide us with a shortcut to calculate things more efficiently without manually counting each possibility. In this problem, the formula to calculate the number of subsets is \( 2^n \), where \( n \) is the total number of elements in the set. Here's a basic breakdown:

• The formula \( 2^n \) comes from the idea that each element in a set can either be included in a subset or not.
• Therefore, for \( n \) elements, there are \( 2^n \) possible combinations, or subsets, because each element has two choices: to be included or excluded.

When we applied this to our problem, we set \( n = 9 \) (since there are 9 elements), and calculated \( 2^9 = 512 \). This means there are 512 different ways to form subsets from our set of two-digit numbers that contain the digit '0'. Mathematical formulas like this one simplify and speed up problem-solving processes in math.