Question

Draw a Venn diagram of the sets described. Of the positive integers less than 15 , set \(A\) consists of the factors of 15 and set \(B\) consists of all odd numbers.

Short answer

The factors of 15, which are 1, 3, 5, and 15 should be placed in set \(A\) in the Venn diagram, while the set \(B\) should contain all the odd numbers less than 15, which are 1, 3, 5, 7, 9, 11, and 13. In the overlapping area of the two circles in the Venn diagram (the intersection of the two sets), it should contain numbers that are both factors of 15 and which are odd numbers less than 15, namely 1, 3, and 5.

Step-by-step solution

- Find the factors of 15

Firstly, it's needed to find the factors of 15 to define set \(A\). Factors of 15 are the integers that we can evenly divide into 15. We can pair them in multiplications thatresult in 15, like \(1 \times 15\), or \(3 \times 5\). Therefore, the factors of 15 are: 1, 3, 5, and 15.

- Identify the odd numbers less than 15

Set \(B\) is defined as all odd numbers less than 15. Oddnumbers are integers that are not divisible by 2. Here are the odd numbers lesser than 15: 1, 3, 5, 7, 9, 11, and 13.

- Draw a Venn diagram

It's needed to create two circles that overlap. One is for set \(A\) and the other one is for set \(B\). In set \(A\), include all the factors of 15 (1, 3, 5, and 15) and in set \(B\), include all the odd numbers less than 15 (1, 3, 5, 7, 9, 11, and 13). By doing this, it's clear that the numbers 1, 3, and 5 appears in both the circles. These numbers are the intersection of sets \(A\) and \(B\) hence should be placed in the overlapped area.

Key concepts

Set Theory

Set theory is a branch of mathematical logic that studies sets, which are collections of objects. Sets can contain numbers, variables, or other elements, grouped together based on a specific rule or property they share. In the context of Venn diagrams, sets are visually represented as circles. These circles might overlap, allowing us to easily observe intersections and differences between the sets.
A set can be described in several ways, such as listing its elements or defining its properties. For example, set \( A \) in our exercise refers to the factors of 15, while set \( B \) lists odd numbers less than 15.

• The union of two sets encompasses all elements from both sets.
• The intersection consists of elements that are common to both sets.
• The difference highlights elements in one set but not in the other.

Understanding these operations helps in constructing and interpreting Venn diagrams effectively. For our Venn diagram in the exercise, the overlap between sets \( A \) and \( B \) includes the numbers shared by both, which are 1, 3, and 5.

Factors and Multiples

The concept of factors and multiples is fundamental in mathematics. Factors of a number are the integers that divide that number exactly without leaving any remainder. For example, the factors of 15 are those numbers which, when multiplied in pairs, produce 15:

• \(1 \times 15 = 15\)
• \(3 \times 5 = 15\)

Hence, the factors of 15 are 1, 3, 5, and 15.
On the other hand, multiples of a number are the product of that number and any integer. For example, the multiples of 3 include 3, 6, 9, 12, and continue indefinitely. Recognizing these distinctions between factors and multiples is crucial for tasks like identifying set \( A \) in our exercise.

Odd Numbers

Odd numbers are integers that cannot be divided evenly by 2. When divided by 2, these numbers leave a remainder of 1. Understanding this property helps in identifying which numbers belong to a set based on their odd nature.
In our exercise, set \( B \) consists of all odd numbers less than 15, forming an easy-to-identify sequence: 1, 3, 5, 7, 9, 11, and 13. These numbers exhibit regularity and can be quickly listed by starting at 1 and adding 2 successively.
The distinction of odd numbers is useful in many mathematical contexts, including their role in identifying intersections within a Venn diagram. By recognizing which odd numbers are factors of a number like 15, we can easily find the overlap with other sets. In our case, numbers 1, 3, and 5 fall into both categories, helping form the intersection of sets \( A \) and \( B \).