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Chapter 0: Problem 34

Find the union of the sets. $$\\{e, m, p, t, y\\} \cup \varnothing$$

Short Answer

Expert verified
The union of the sets \(\{e, m, p, t, y\}\) and \(\varnothing\) is \(\{e, m, p, t, y\}\).

Step by step solution

01

Understand Set Union

In set theory, the union of two sets is a set containing all elements of both sets. The symbol for union is \(\cup\). If \(A\) and \(B\) are two sets, then their union is given by \(A \cup B\).
02

Union with Empty Set

The union of any set with the empty set (\(\varnothing\)) is just the original set itself. This is because the empty set has no elements to add to the original set. Thus, for any set \(A\), we have \(A \cup \varnothing = A\).
03

Apply Rule to Given Sets

In this problem, your set \(A\) is \(\{e, m, p, t, y\}\). Using the rule from step 2, you can find that the union of \(A\) and the empty set is \(\{e, m, p, t, y\} \cup \varnothing = \{e, m, p, t, y\}\).

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Key Concepts

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

Set Union
Set union is a fundamental concept in set theory, which deals with the combination of elements from two sets. When you take the union of two sets, you combine all unique elements from both sets into a new set. This means that any element found in either of the sets A or B will be included in the union set. The union operation is essential for understanding how different sets relate to each other in mathematics and everyday scenarios.
Some important points about set union include:
  • The result of a union is always a set with all unique elements; duplicate elements are not repeated more than once.
  • The union of any set with itself, for example, A 鈭� A, results in the same set: A.
Set union is not only used in mathematics but also in various fields like computer science, where data sets are merged to analyze and interpret data.
Empty Set
An empty set, denoted by \( \varnothing \) or sometimes by \( \{ \} \), is a set that contains no elements. This special set is unique in set theory and plays an essential role when discussing operations like union or intersection. The empty set is the identity element for set union, meaning that when you take the union of any set with the empty set, the result is the original set itself.
Here are some cool facts about the empty set:
  • The empty set is the only set with zero cardinality, meaning it has no elements to count.
  • It is a subset of every set, which means every set includes the empty set as one of its subsets.
Understanding the empty set is vital for foundational set theory operations and for grasping concepts of nullity and absence in mathematical logic.
Union Symbol
The union symbol \( \cup \) is a straightforward and intuitive notation used in set theory to represent the union of two or more sets. When you see this symbol between two sets, think of it as a simple instruction to "combine everything you see into one set." This symbol helps avoid confusion and makes it easier to express operations involving sets in a clear, concise manner.
Here's a quick guide to using the union symbol:
  • When writing set expressions, the union symbol goes between the sets to be combined, like \( A \cup B \).
  • A common mistake is confusing union \( \cup \) with intersection \( \cap \); remember, union is about "everything together," while intersection focuses on "common elements only."
Mastering the use of the union symbol and distinguishing it from other similar symbols is vital for anyone delving into set theory and related mathematical disciplines.

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Most popular questions from this chapter

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. Parts for an automobile repair cost \(\$ 175 .\) The mechanic charges \(\$ 34\) per hour. If you receive an estimate for at least \(\$ 226\) and at most \(\$ 294\) for fixing the car. what is the time interval that the mechanic will be working on the job?

Explain how to solve \(x^{2}+6 x+8=0\) using the quadratic formula.

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If you are given a quadratic equation, how do you determine which method to use to solve it?
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