Is R A Subset Of A Given U = {x | X Is A Real Number}, A = {x | X Is An Odd Integer}, And R = {x | X = 3, 7, 11, 27}?

In the realm of mathematics, set theory provides a fundamental framework for understanding collections of objects. Sets are well-defined collections, and the relationships between them, such as subsets, are crucial in various mathematical disciplines. In this article, we will delve into the concept of subsets and analyze the specific case of determining whether a set R is a subset of set A. We will be exploring the definitions of the sets U, A, and R and rigorously examining whether all elements of R are also elements of A. This involves understanding the properties of real numbers, odd integers, and specific numerical values. The conclusion will definitively answer the question of whether R is indeed a subset of A, solidifying our understanding of set relationships.

Defining the Sets: U, A, and R

Before we can determine if R is a subset of A, we must first understand the composition of each set.

• The Universal Set (U): The universal set, denoted by U, encompasses all elements under consideration in a particular context. In this case, U is defined as the set of all real numbers. Real numbers constitute a vast number system that includes rational numbers (which can be expressed as a fraction p/q, where p and q are integers and q is not zero) and irrational numbers (which cannot be expressed as a fraction, such as √2 or π). U acts as the overarching domain within which our other sets reside. It's the all-encompassing set from which we draw our elements.

• Set A: The Set of Odd Integers: Set A is defined as the set of all odd integers. Integers are whole numbers (without any fractional or decimal components) and can be positive, negative, or zero. Odd integers, specifically, are integers that are not divisible by 2. They can be expressed in the form 2n + 1, where n is any integer. Examples of odd integers include -3, -1, 1, 3, 5, and so on. Understanding this definition is crucial because it sets the criteria for membership in A. An element must be an integer and must leave a remainder of 1 when divided by 2 to qualify.

• Set R: A Specific Set of Numbers: Set R is explicitly defined as containing the numbers 3, 7, 11, and 27. These are specific numerical values, and our task is to determine if these specific numbers also belong to set A. The limited number of elements in R makes it straightforward to examine each element individually against the criteria for membership in A. Each of these numbers is a positive integer. We need to ascertain if each of these integers also meets the condition of being an odd integer.

Understanding Subsets: The Key to the Question

The core question at hand is whether R is a subset of A. To answer this, we must understand the definition of a subset. A set R is considered a subset of a set A if every element in R is also an element in A. In mathematical notation, this is represented as R ⊆ A. Conversely, if there exists at least one element in R that is not in A, then R is not a subset of A. This definition provides us with a clear criterion to evaluate the relationship between R and A. We must systematically check each element of R for membership in A.

Applying the Subset Definition to Our Sets

To determine if R ⊆ A, we must verify that each element in R (3, 7, 11, and 27) is also an element in A. This requires us to check if each number is an odd integer. We know from the definition of set A that its members are odd integers. Therefore, if all elements of R fit this description, then R is a subset of A. If even one element of R does not meet this criterion, then R is not a subset of A.

Analyzing the Elements of Set R

Now, let's examine each element of R individually to see if it qualifies as an odd integer:

• 3: The number 3 is an integer. When 3 is divided by 2, it leaves a remainder of 1. Thus, 3 is an odd integer. Therefore, 3 belongs to set A.

• 7: Similarly, 7 is an integer. Dividing 7 by 2 yields a remainder of 1, confirming that 7 is an odd integer. Therefore, 7 is also a member of set A.

• 11: The number 11 is an integer as well. Dividing 11 by 2 leaves a remainder of 1, indicating that 11 is an odd integer. Hence, 11 belongs to set A.

• 27: Lastly, 27 is an integer. When 27 is divided by 2, the remainder is 1. This confirms that 27 is an odd integer and is, therefore, an element of set A.

Synthesizing the Element Analysis

We have individually confirmed that each element of R (3, 7, 11, and 27) meets the criteria for membership in set A. Each number is an odd integer. This comprehensive analysis is crucial because the definition of a subset requires that all elements of the potential subset must belong to the larger set. Finding even a single element that does not belong to the larger set would invalidate the subset relationship.

Determining if R is a Subset of A: The Conclusion

Having meticulously analyzed each element of set R and confirmed that they are all odd integers, we can now definitively answer the question: Is R ⊆ A? The answer is yes.

Justification for the Conclusion

Our conclusion is based on the fundamental definition of a subset. A set R is a subset of set A if and only if every element in R is also an element in A. We have systematically verified that 3, 7, 11, and 27—the elements of R—are all odd integers, which is the defining characteristic of set A. Therefore, all elements of R are indeed present in A. There are no exceptions or counterexamples. This direct application of the subset definition leads us to our affirmative conclusion.

Final Answer

Therefore, with a strong understanding of set theory and careful examination of the elements involved, we can confidently conclude that yes, R is a subset of A. This underscores the importance of precise definitions and rigorous analysis in mathematical problem-solving. This detailed exploration of set relationships provides a clear illustration of how mathematical concepts are applied to specific scenarios, enhancing our understanding of set theory and its applications.