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Chapter 6: Problem 18

Let \(A=\\{a, e, l, t, r\\} .\) Which of the following sets are equal to \(A\) ? a. \(\\{x \mid x\) is a letter of the word later \(]\) b. \(\\{x \mid x\) is a letter of the word latter\\} c. \(\\{x \mid x\) is a letter of the word relate \(\\}\)

Short Answer

Expert verified
Sets a, b, and c are all equal to Set A.

Step by step solution

01

Identify Set A

Set A is the set containing the elements: \(\\{a, e, l, t, r\\}\).
02

Identify Set a

Set a includes the elements from the word "later". We list them down in a set while ensuring that each letter is included only once: \(\\{l, a, t, e, r\\}\).
03

Identify Set b

Set b includes the elements from the word "latter". We list them down in a set, again ensuring that each letter is included only once: \(\\{l, a, t, e, r\\}\).
04

Identify Set c

Set c includes the elements from the word "relate". We list them down in a set, again ensuring that each letter is included only once: \(\\{r, e, l, a, t\\}\).
05

Comparing Sets

We now compare each identified set with Set A to determine which sets, if any, are equal. - Set a: \(\\{l, a, t, e, r\\}\) has the same elements as Set A in a different order; therefore, Set a is equal to Set A. - Set b: \(\\{l, a, t, e, r\\}\) also has the same elements as Set A in a different order; therefore, Set b is equal to Set A. - Set c: \(\\{r, e, l, a, t\\}\) has the same elements as Set A in a different order; therefore, Set c is equal to Set A. In conclusion, all the given sets - a, b, and c - are equal to Set A.

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Key Concepts

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

Equality of Sets
In set theory, two sets are deemed equal if and only if they have precisely the same elements. This concept is fundamental as it implies that order and repetition do not matter in a set.
For example, the set \[ \{a, b, c\} \] is considered equal to \[ \{c, a, b\} \] since they contain the same elements, albeit ordered differently.
When determining the equality of sets, it is crucial to ensure all elements present in one set are found equivalently in the other. This is what we applied to decide that sets a, b, and c in the original exercise are each equal to set A.
Elements of a Set
Elements or members are the individual components that make up a set. In set notation, each element gets listed within curly braces separated by commas.
For instance, if we have a set denoted as: \( B = \{x, y, z\} \), then x, y, and z are the elements of set B.
In the exercise, identifying the elements of various sets derived from different words allowed us to evaluate equality. Each set was built from the letters of words "later," "latter," and "relate," ensuring each letter appeared only once, leading to the formations of distinct sets for comparison.
Understanding elements is key to manipulating and comparing sets effectively.
Set Notation
Set notation provides a clear and concise way to describe the elements of a set. Sets are typically expressed in curly braces, with elements separated by commas.
For example, the set containing numbers 1, 2, and 3 is written as: \(\{1, 2, 3\}\).
In the exercise, each set was a collection of letters, showing how to use set notation with non-numeric elements. This helps organize and communicate the specifics of a set effectively. Moreover, the use of notation like \(\{x \mid x \text{ is a letter of the word ...}\}\) allows for a formal description of the set's members.
With this foundation, it becomes easier to make logical and precise comparisons between different sets.
Finite Sets
Finite sets are sets that contain a countable number of elements. This is in contrast to infinite sets, which have no bounds on the number they can include.
For example, \( \{2, 4, 6, 8\} \) is a finite set with exactly four elements.
In the original exercise, the sets derived from the words "later," "latter," and "relate" are finite since the alphabet's letters limit them, and each word provides a specific number of letters to form a set.
Understanding finite sets is important as it underscores the limits and structure within which these sets operate, aiding both in their comprehension and application in problems.

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Most popular questions from this chapter

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