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Chapter 6: Problem 9

State whether the statements are true or false. $$ \text { a. }\\{a, b, c\\}=\\{c, a, b\\} $$

Short Answer

Expert verified
True. The sets \(\{a, b, c\}\) and \(\{c, a, b\}\) are equal because they have the same elements, regardless of their order.

Step by step solution

01

Identify the Elements of Both Sets

First, we need to identify the elements in both sets. - Set A: \(\{a, b, c\}\) - Set B: \(\{c, a, b\}\)
02

Compare the Elements of Both Sets

Now, we will compare the elements of both sets. - Set A has elements: a, b, and c. - Set B has elements: c, a, and b. Looking at both sets, we can see that they have the same elements: a, b, and c.
03

Determine if the Sets are Equal

Since both sets have the same elements, regardless of their order, we can conclude that \(\\{a, b, c\\}=\\{c, a, b\\}\). So, the statement is true.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Elements of a Set
When we consider elements of a set, we are referring to the distinct objects that make up that set. In set theory, an element is something that makes up the collective group known as a set and can be anything: numbers, letters, objects, or even other sets. An essential aspect of sets is that the order in which elements are listed does not matter, nor does any repetition of elements affect the set.

For example, in the given exercise, we have two sets: Set A \(\{a, b, c\}\) and Set B \(\{c, a, b\}\). The order of the elements is different, yet in set theory, this does not make the sets distinct. What matters is solely the presence of the elements, which in both cases are the same. This concept can sometimes trip students up, but remembering that a set is defined by what is in it, not how it is presented, can clear up much confusion.
Comparing Sets
The process of comparing sets is a cornerstone in set theory and involves looking at the elements within each set to determine a relationship. In terms of equality, two sets are considered equal if they contain exactly the same elements, with no more and no less. This is also known as the axiom of extensionality, which states that a set is completely determined by its elements.

When comparing sets like Set A and Set B in the provided exercise, we simply look to see if one set contains all the elements of the other and vice versa. The sequence of the elements does not matter when determining equality. Therefore, when we compare \(\{a, b, c\}\) and \(\{c, a, b\}\), despite their different arrangements, they are equal because the collections of elements are identical.
Properties of Sets
Exploring properties of sets furthers our understanding of how sets work. Here are a few foundational properties:
  • Uniqueness: Every element in a set is unique. Duplicate entries are not counted separately.
  • Unordered: The arrangement of elements in a set does not matter. Sets \(\{a, b, c\}\) and \(\{b, c, a\}\) are the same because they contain the same elements, irrespective of order.
  • Well-defined: A set is well-defined if its membership is clear and unambiguous. This means that it is always possible to determine whether an object is an element of a set or not.
These properties play a critical role when we consider the equality of two sets. As we've seen in the exercise, because both sets \(\{a, b, c\}\) and \(\{c, a, b\}\) have the same elements, without repetition, and order does not matter, we conclude that they are equal following these properties of sets.

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