Set Operations Explained Finding Union And Intersection Of Sets A, B, And C
In the realm of mathematics, particularly in set theory, understanding set operations is crucial. Set operations allow us to combine, compare, and analyze different sets, providing a foundation for more advanced mathematical concepts. This article delves into three fundamental set operations: union (∪), intersection (∩), and their combination. We will explore these operations using specific examples involving sets A, B, and C, which include intervals, sets of natural numbers, and sets of real numbers. Our goal is to provide a clear and comprehensive understanding of how these operations work and how they can be applied.
Problem Statement
Given three sets:
- A = (-∞, 10] (all real numbers less than or equal to 10)
- B = {x | -6 < x ≤ 11, x ∈ N} (all natural numbers greater than -6 and less than or equal to 11)
- C = {x | -2 < x < 8, x ∈ R} (all real numbers greater than -2 and less than 8)
We aim to find:
- a) A ∪ B (the union of sets A and B)
- b) B ∩ C (the intersection of sets B and C)
- c) A ∩ (B ∪ C) (the intersection of set A with the union of sets B and C)
a) Finding A ∪ B (The Union of A and B)
The union of two sets, denoted by A ∪ B, is a set that contains all the elements that are in A, or in B, or in both. In simpler terms, we combine all the elements from both sets into a single set, without duplicating any element. To find A ∪ B, we need to consider the elements present in set A and set B.
Set A: (-∞, 10]
Set A includes all real numbers from negative infinity up to and including 10. This means any number less than or equal to 10 belongs to set A. Examples of elements in set A include -100, -5, 0, 5, 10, and any real number in between.
Set B: {x | -6 < x ≤ 11, x ∈ N}
Set B consists of natural numbers (positive integers) that are greater than -6 and less than or equal to 11. Natural numbers are positive integers, so we consider the integers 1, 2, 3, and so on. Therefore, the elements of set B are {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}.
Combining A and B
To find A ∪ B, we combine all elements from set A and set B. Since set A includes all real numbers up to 10, it already encompasses many elements in set B. The only element in set B that is not included in the interval of set A is 11. Therefore, A ∪ B includes all real numbers less than or equal to 10, along with the number 11.
A ∪ B = (-∞, 10] ∪ {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11} = (-∞, 10] ∪ {11}
In interval notation, this can be represented as (-∞, 10] ∪ {11}. This means the set includes all real numbers up to 10 and the single element 11. Understanding the union operation is crucial, as it forms the basis for many set theory applications in mathematics and computer science.
b) Finding B ∩ C (The Intersection of B and C)
The intersection of two sets, denoted by B ∩ C, is a set that contains only the elements that are common to both sets B and C. In other words, an element must be present in both set B and set C to be included in their intersection. To determine B ∩ C, we need to identify the elements shared between set B and set C.
Set B: {x | -6 < x ≤ 11, x ∈ N}
As we previously established, set B consists of natural numbers greater than -6 and less than or equal to 11. The elements of set B are {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}.
Set C: {x | -2 < x < 8, x ∈ R}
Set C includes all real numbers greater than -2 and less than 8. This means any number between -2 and 8 (excluding -2 and 8) belongs to set C. Examples of elements in set C include -1.5, 0, 2.5, 5, and 7.99.
Identifying Common Elements
To find B ∩ C, we look for elements that are present in both set B and set C. Set B contains natural numbers, while set C contains real numbers within a specific range. The natural numbers that fall within the range of set C (-2 < x < 8) are 1, 2, 3, 4, 5, 6, and 7. These are the only elements that are common to both sets.
B ∩ C = {1, 2, 3, 4, 5, 6, 7}
The intersection of set B and set C is the set {1, 2, 3, 4, 5, 6, 7}. This demonstrates how the intersection operation helps in identifying commonalities between sets, a fundamental concept in database queries, logic, and various other fields.
c) Finding A ∩ (B ∪ C) (The Intersection of A with the Union of B and C)
To find A ∩ (B ∪ C), we first need to determine the union of sets B and C (B ∪ C), and then find the intersection of the resulting set with set A. This involves combining two set operations, making it a more complex but essential concept in set theory. Breaking down the problem into steps helps in understanding the process more clearly.
Step 1: Find B ∪ C (The Union of B and C)
As previously discussed, the union of two sets is a set that contains all the elements that are in either set. We need to combine the elements of set B and set C.
- Set B: {x | -6 < x ≤ 11, x ∈ N} = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}
- Set C: {x | -2 < x < 8, x ∈ R}
When combining these sets, we include all the elements of set B and set C. Set C contains all real numbers between -2 and 8. Set B contains natural numbers up to 11. Therefore, B ∪ C will include all real numbers between -2 and 8, along with the natural numbers 8, 9, 10, and 11 from set B (since they are not already included in set C).
B ∪ C = (-2, 8) ∪ {8, 9, 10, 11} = (-2, 11]
In interval notation, B ∪ C is represented as (-2, 8) ∪ {8, 9, 10, 11}. However, since 8 is a boundary element of set C, and we include the integers 8 to 11 from set B, we can simplify the notation to (-2, 8) combined with the discrete values 8, 9, 10, and 11. A simpler notation for the union is (-2, 11], representing all real numbers greater than -2 and less than or equal to 11.
Step 2: Find A ∩ (B ∪ C) (The Intersection of A with B ∪ C)
Now that we have B ∪ C, we need to find the intersection of this set with set A.
- Set A: (-∞, 10] (all real numbers less than or equal to 10)
- B ∪ C: (-2, 11] (all real numbers greater than -2 and less than or equal to 11)
The intersection of A and (B ∪ C) will contain elements that are common to both sets. Set A includes all real numbers up to 10, while B ∪ C includes real numbers greater than -2 and up to 11. The common elements are those greater than -2 and less than or equal to 10.
A ∩ (B ∪ C) = (-∞, 10] ∩ (-2, 11] = (-2, 10]
Therefore, A ∩ (B ∪ C) is the interval (-2, 10], which includes all real numbers greater than -2 and less than or equal to 10. This intersection operation, combined with the union operation, provides a comprehensive way to manipulate and understand sets, which is vital in mathematical analysis and computer science algorithms.
In summary, we have explored the set operations of union (∪) and intersection (∩) using three given sets: A, B, and C. We found that:
- A ∪ B = (-∞, 10] ∪ {11}
- B ∩ C = {1, 2, 3, 4, 5, 6, 7}
- A ∩ (B ∪ C) = (-2, 10]
Understanding these operations is fundamental in set theory and has wide-ranging applications in various fields, including mathematics, computer science, and data analysis. By mastering these concepts, one can better analyze and solve problems involving sets and their relationships.
This exploration provides a solid foundation for further study in set theory and its applications. The union and intersection operations are building blocks for more complex concepts, such as set differences, complements, and Cartesian products, which are essential in advanced mathematical studies and practical applications.
