Problem 17 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Let \(A=\\{1,2,3\\}\). Which of the following sets are equal to \(A\) ? a. \(\\{2,1,3\\}\) b. \(\\{3,2,1\\}\) c. \(\\{0,1,2,3\\}\)

Step-by-Step Solution

Answer

Sets a (\{2, 1, 3\}) and b (\{3, 2, 1\}) are equal to set A (\{1, 2, 3\}), while set c (\{0, 1, 2, 3\}) is not.

Step 1: Option a: Set {2, 1, 3}

Comparing set A and this option, we can see that they both consist of elements 1, 2, and 3, despite the order being different. Therefore, this set is equal to A.

Step 2: Option b: Set {3, 2, 1}

Comparing set A and this option, we can see that they both consist of elements 1, 2, and 3, the order is reversed, but as we know, the order does not matter in sets. Therefore, this set is equal to A.

Step 3: Option c: Set {0, 1, 2, 3}

Comparing set A and this option, we can see that this set contains one additional element, 0, which is not present in set A. Therefore, this set is not equal to A. To conclude, sets a and b are equal to set A, while set c is not.

Key Concepts

Properties of Sets

Set theory, the foundation of modern mathematics, involves the study of collections of objects known as 'sets'. Understanding the properties of sets is crucial in determining how sets relate to one another. One key property is that the order of elements in a set does not matter. This means that \( \{2, 1, 3\} \) is considered identical to \( \{1, 2, 3\} \) because they contain the same elements irrespective of the order. Another fundamental property is that sets do not contain duplicate elements. Hence, \( \{1, 1, 2, 3\} \) simplifies to \( \{1, 2, 3\} \). Additionally, the concept of subset and superset further explore the relationships between sets, where one set may contain all the elements of another set.

These properties facilitate the comparison of sets, assisting in exercises such as identifying equal sets. In the given problem, recognizing that the order of elements does not affect the identity of a set is crucial for correctly determining that Option a and Option b represent the same set as \(A\). Knowing that sets contain distinct elements helps identify that Option c, with an additional '0' element, is not equal to \(A\).

Mathematical Sets

In mathematics, a 'set' is a well-defined collection of distinct objects, considered as an object in its own right. Sets are typically denoted by curly braces, such as \( \{1, 2, 3\} \). Objects within the set are called 'elements' or 'members'. The nature of these elements can vary—they could be numbers, letters, symbols, or even other sets, highlighting the versatility of set theory in different areas of mathematics.

When engaging with mathematical sets, it's important to be precise about what constitutes a member of the set to avoid ambiguity. This precision was underscored in the problem provided, where clarity about set membership was crucial for determining equality of sets. In practice, this involves laying out the 'elements' within the brackets, ensuring that each element is distinct and properly accounted for. In educational contexts, illustrating concrete examples like those in the provided exercise are invaluable for reinforcing students' understanding of set concepts.

Comparing Sets

Comparing sets is a fundamental part of set theory that involves analyzing the elements of sets to understand their relationship. Two sets are considered equal if they contain the exact same elements with no regard to the order. As observed in the exercise, even if the arrangement of numbers is different, as long as the members are identical, the sets are equal.

Another aspect of comparing sets is the consideration of subsets and supersets. A subset is a set whose elements are all contained within another set, while a superset has all the elements of a subset plus possibly more. In the context of the exercise, Option c is a superset of set \(A\) because it contains all elements of \(A\), in addition to the element '0'. This clarity helps in quickly identifying unequal sets.

• If two sets have the exact same elements, they are equal.
• If a set contains elements that are not present in the other set, they are not equal.

Understanding these principles is indispensable for students tackling set comparisons and is highlighted through practical exercises like the one demonstrated.