Question

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. If \(A=\\{2,4,6, \ldots, 30\\} \quad\) and \(\quad B=\\{3,6,9, \ldots, 30\\}\) find \(A \cap B\)

Short answer

\( A \cap B = \{ 6, 12, 18, 24, 30 \} \)

Step-by-step solution

- Understand Set A

Set A contains the even numbers between 2 and 30. We can write it as follows: \[ A = \{2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30\} \]

- Understand Set B

Set B contains the multiples of 3 between 3 and 30. We can write it as follows: \[ B = \{3, 6, 9, 12, 15, 18, 21, 24, 27, 30\} \]

- Find Common Elements (Intersection)

To find the intersection of sets A and B (denoted as \( A \cap B \)), identify the common elements in both sets. Compare the elements of sets A and B: Common elements are 6, 12, 18, 24, and 30.

- Write the Intersection

Write the common elements found in step 3 as the intersection of sets A and B. Thus, \[ A \cap B = \{ 6, 12, 18, 24, 30\} \]

Key concepts

Sets in Mathematics

In mathematics, a set is a collection of distinct objects, considered as an object in its own right. For example, the numbers 2, 4, and 6 can be considered as elements of a set. Sets are usually denoted by curly braces \(\{ \}\) and the elements inside are listed, separated by commas. For instance, the set containing 1, 2, and 3 is written as \(\{1, 2, 3\}\).

Sets can represent a variety of things. Some sets are finite, containing a specific number of elements. Others are infinite, with an infinite number of elements. Set theory is fundamental to mathematics because it forms the building blocks for more complex concepts.

When working with sets, we may perform operations such as union (\(\cup\)), intersection (\(\cap\)), and difference (\( - \)).

• Union (\(\cup\)): Combines all elements from both sets.
• Intersection (\(\cap\)): Contains only elements common to both sets.
• Difference (\( - \)): Elements in one set but not in the other.

Let's move on to a practical example involving sets.

Even Numbers

An even number is an integer that is exactly divisible by 2. In other words, when an even number is divided by 2, there is no remainder. Examples of even numbers are 2, 4, 6, 8, and so on.

In set notation, if we were to list the even numbers between 2 and 30, it would look like this: \(\{2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30\}\). Notice that each number in this set follows the rule of divisibility by 2.

Understanding even numbers is crucial for comprehending various mathematical concepts and problems, including set operations. These numbers come into play in different types of numerical sequences and patterns we may encounter.

Multiples of Numbers

Multiples of a number are the results of multiplying that number by integers. For example, the multiples of 3 include 3, 6, 9, 12, and so on. To get these multiples, you simply multiply 3 by 1, 2, 3, etc. Mathematically, a multiple of 3 can be represented as \(3n\), where \(n\) is any integer.

If we want to list the multiples of 3 between 3 and 30, we write: \(\{3, 6, 9, 12, 15, 18, 21, 24, 27, 30\}\). These numbers are all divisible by 3 without leaving a remainder.

Understanding multiples is fundamental in various arithmetic operations and can help in finding common factors. When we look at intersections of sets, knowing the multiples can help identify shared elements quickly. In our exercise example, creating a set of multiples and another set of even numbers allows us to find elements that appear in both sets.