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Chapter 4: Problem 22

Find each indicated intersection or union. $$ \\{u, v, w\\} \cup\\{u, w\\} $$

Short Answer

Expert verified
The union is \( \{u, v, w\} \).

Step by step solution

01

Understand the Union Function

The union of two sets is a set containing all elements from both sets. If elements are repeated in both sets, they are only included once in the union.
02

Identify the Elements in Each Set

The elements in the first set are \( \{u, v, w\} \) and the elements in the second set are \( \{u, w\} \).
03

Combine Elements From Both Sets

Combine the elements from both sets making sure to include each element only once. So, \( \{u, v, w\} \cup \{u, w\} \) includes the elements \( u, v, \) and \( w \).

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Key Concepts

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

Union of Sets
When we talk about the 'union of sets', we mean combining all the elements from two or more sets into one set. This is represented by the symbol \(\bigcup\).
For instance, given two sets A = \(\{u, v, w\}\) and B = \(\{u, w\}\), the union A \(\bigcup\) B would include all the elements from both sets without any repetition.
It's important to remember that even if an element appears in both sets, it should only appear once in the union. So, for our example, A \(\bigcup\) B results in \(\{u, v, w\}\). Here are some key points:
  • Union gathers all distinct elements from the sets involved.
  • Repeated elements are counted only once.
  • The order of elements does not matter.
Elements
In set theory, 'elements' refer to the distinct objects or members belonging to a set.
For example, the set \(\{u, v, w\}\) contains the elements 'u', 'v', and 'w'. These elements are distinct items that make up the set.
Each element should be unique within its set. Here are some key points about elements:
  • Distinct items make up a set.
  • An element can be any identifiable object: a number, letter, symbol, etc.
  • In the context of sets, elements are not repeated.
Repetition in Sets
An essential concept in set theory is understanding 'repetition in sets'. In sets, an element can only appear once, even if it is mentioned multiple times in original sets. This principle avoids redundancy and maintains uniqueness among the elements.
To illustrate, consider the sets A = \(\{u, v, w\}\) and B = \(\{u, w\}\). When taking the union A \(\bigcup\) B, element 'u' and 'w' will be included only once in the result, despite appearing in both sets.
Here are key points regarding repetition in sets:
  • Each element appears only once in a set.
  • Union operations do not repeat common elements.
  • Sets avoid redundancy and emphasize distinctness.

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Most popular questions from this chapter

Age of Marriage. The following rule of thumb for determining an appropriate difference in age between a bride and a groom appears in many Internet blogs: The younger spouse's age should be at least seven more than half the age of the older spouse. Let \(b=\) the age of the bride, in years, and \(g=\) the age of the groom, in years. Write and graph a system of inequalities that represents all combinations of ages that follow this rule of thumb. Should a minimum or maximum age for marriage exist? How would the graph of the system of inequalities change with such a requirement?

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