Problem 26 - Algebra for College Students

Question

Use the following set designations. $$ \begin{aligned} &N=\\{x \mid x \text { is a natural number }\\} \\ &Q=\\{x \mid x \text { is a rational number }\\} \\ &W=\\{x \mid x \text { is a whole number }\\} \\ &H=\\{x \mid x \text { is an irrational number }\\} \\ &I=\\{x \mid x \text { is an integer }\\} \\ &R=\\{x \mid x \text { is a real number }\\} \end{aligned} $$ Place \(\subseteq\) or \(\nsubseteq\) in each blank to make a true statement. \(W\)_____ \(I\)

Step-by-Step Solution

Answer

\( W \subseteq I \)

Step 1: Define Whole Numbers (W)

The set of whole numbers, denoted as \( W \), includes all non-negative integers: \( \{0, 1, 2, 3, ext{...} \} \).

Step 2: Define Integers (I)

The set of integers, denoted as \( I \), includes all whole numbers and their negative counterparts: \( \{..., -3, -2, -1, 0, 1, 2, 3, ...\} \).

Step 3: Determine Set Relationship

A set \( A \) is a subset of set \( B \) (\( A \subseteq B \)) if every element of \( A \) is also an element of \( B \). Observe that all elements of \( W \) are found in \( I \).

Step 4: Final Comparison and Conclusion

Since every whole number is also an integer, we can conclude \( W \subseteq I \).

Key Concepts

whole numbers

Whole numbers form a basic set in mathematics and include zero together with all positive integers. Essentially, whole numbers are just the set of natural numbers plus zero:

• Numbers like 0, 1, 2, 3, and so on are whole numbers.
• They do not include negative numbers or fractions.

The concept of whole numbers is fundamental in arithmetic, as it signifies all complete units. Understanding this set is important because it functions as a building block in mathematical problem-solving. Breaking up larger sets into natural and whole numbers helps with grasping concepts, like identifying subsets in set theory easily. For instance, when resolving questions about subsets, you should identify if given numbers align with features of whole numbers. Knowing that whole numbers are non-negative integers makes recognizing set inclusion easier, since this entire set falls perfectly into the set of integers.

integers

Integers are a more extensive class of numbers when compared to whole numbers. This set includes both negative and positive numbers, along with zero. More precisely, the set of integers can be represented as:

• Negative numbers: ...-3, -2, -1
• Zero: 0
• Positive numbers: 1, 2, 3, ...

Integers are crucial in mathematics because they include negatives, positives, and zero, which means they essentially envelope whole numbers. When considering integers in the context of subset questions, it’s all about checking if the smaller set (like whole numbers) fits entirely within the larger set (integers). This characteristic is vital for distinguishing relationships in sets, as complete elements of whole numbers align comfortably within the set of integers.

subsets

The concept of subsets is central to understanding set theory.If every element of set A exists in another set B, then set A is a subset of set B, denoted as \( A \subseteq B \). This relationship is essential in exercises and proofs because it helps establish inclusions and hierarchies between sets. When determining if one set is a subset of another:

• Check each element in the first set.
• Verify they all are elements within the second set.

For instance, the set of whole numbers \( W \) is indeed a subset of the set of integers \( I \). Each whole number, like 0 or 5, is found in the integer set, thus proving the subset relationship \( W \subseteq I \).Comprehending this notion aids in predicting and verifying which sets fit into others, forming foundational knowledge for solving mathematical relations.