Problem 18 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

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 \(\\}\)

Step-by-Step Solution

Answer

Sets a, b, and c are equal to set A, as they contain exactly the same elements as set A: \(\\{a, e, l, t, r\\}\).

Step 1: Elements in Set A

\(A = \\{a, e, l, t, r\\}\) Set a:

Step 2: Elements in Set a

\(a = \\{x \mid x \text{ is a letter of the word later}\\} = \\{a, e, l, t, r\}\\) Set b:

Step 3: Elements in Set b

\(b = \\{x \mid x \text{ is a letter of the word latter}\\} = \\{a, e, l, t, r\}\\) Set c:

Step 4: Elements in Set c

\(c = \\{x \mid x \text{ is a letter of the word relate}\\} = \\{a, e, l, t, r\}\\)

Step 5: Compare Sets

Now, let's compare each set to set A: 1. Set a has the same elements as set A: \(\\{a, e, l, t, r\\}\) 2. Set b has the same elements as set A: \(\\{a, e, l, t, r\\}\) 3. Set c has the same elements as set A: \(\\{a, e, l, t, r\\}\)

Step 6: Conclusion

All the sets a, b, and c are equal to set A as they contain exactly the same elements as set A.

Key Concepts

Equality of Sets

Equality of sets is a foundational concept in set theory. Two sets are considered equal if they have exactly the same elements. The order of the elements does not matter, and duplicate elements are ignored. In simpler terms, if every element in Set A is also in Set B, and every element in Set B is in Set A, then Set A and Set B are equal.

For example, the sets \(\{1, 2, 3\}\) and \(\{3, 2, 1\}\) are equal because they have the same numbers, even though the order is different. In the exercise, Sets a, b, and c are all equal to Set A because they all contain the exact same elements: \(\{a, e, l, t, r\}\).

• Order is irrelevant
• Duplicate elements are ignored
• Each element must appear in both sets

Understanding equality is important for recognizing when two sets represent the same collection of objects.

Element Identification

Identifying elements within a set involves determining all the distinct objects or items that belong to the set. This can be done using a list of items, descriptions in words, or rules in set-builder notation like \(\{x \mid x \text{ is a ...}\}\).

In the given exercise, the elements of each set were determined by recognizing the letters from specific words.

• Set A was derived as \(\{a, e, l, t, r\}\).
• Set a uses the letters from 'later', which gives \(\{a, e, l, t, r\}\).
• Set b uses 'latter', but since duplicate letters are ignored, it is \(\{a, e, l, t, r\}\).
• Set c uses 'relate', similarly resulting in \(\{a, e, l, t, r\}\).

This shows how even if words differ in letters used, sets can be equal if duplicates are ignored and the essential letters are the same.

Mathematical Notation

Mathematical notation is a language used to express concepts clearly and unambiguously. In set theory, notation is crucial for defining sets and demonstrating their properties clearly. Set A, in this exercise, is expressed as \(A = \{a, e, l, t, r\}\). Each set is described with curly braces \(\{ \}\), which denote a collection of objects.

When using set-builder notation, sets are defined by a rule such as \(\{x \mid x \text{ is a letter of the word later}\}\). Here, the symbol \(x \mid x\) is used to specify that \(x\) satisfies the given condition. This way, sets can be quickly defined by properties rather than explicit listing.

• Curly braces denote a set
• Set-builder notation provides a rule-based definition
• Mathematical symbols clarify relationships between elements

Being comfortable with this notation is essential for deeper engagement with mathematics, allowing for efficient communication and understanding of complex ideas in set theory.