# Union & Set Rules - Learn the Basics Now!

• MHB
• TinaSprout
In summary, the union of four different sets can equal just one of the sets, as it contains all elements of those sets. In the given example, the union of four intervals, each containing infinitely many elements, is equal to the interval [-1,1], as all elements within the four intervals are already contained in [-1,1]. This can also be proven using the definition of unions and by considering the sets as single elements within a larger set.
TinaSprout
I am going over some of my notes, and I cannot understand unions, here is the selection I am having trouble with
View attachment 7601

How does the union of four different sets equal just one of the sets? Should the union of 4 sets be the four different sets instead of one.
I am missing something fundamental to unions and intersections.

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TinaSprout said:
I am going over some of my notes, and I cannot understand unions, here is the selection I am having trouble with

How does the union of four different sets equal just one of the sets? Should the union of 4 sets be the four different sets instead of one.
I am missing something fundamental to unions and intersections.

Hi TinaSprout! Welcome to MHB! (Smile)

The union of 2 sets is the set that contains all elements that those 2 sets contain.
So e.g. $\{a,b,c\} \cup \{a,d\}=\{a,b,c,d\}$.
That is the combination of all elements.
Note that the element $a$ that is part of both sets, is included once in the union.The notation $[-1,1]$ means that we have a set of infinitely many elements.
Those are all real numbers between -1 and +1.
The union of $[-1,1] \cup [-\frac 12, \frac 12]$ is $[-1,1]$, because it contains all elements of both sets.
That is, it all real numbers between $-\frac 12$ and $\frac 12$ are contained in the interval $[-1,1]$.

In this case, if you make a drawing of the four intervals, I think it already becomes clearer.
However, we can also strictly use the definition in your first equation. We are asked to prove that
$V_1 \cup V_2 \cup V_3 \cup V_4 = [-1,1], \qquad (\ast)$
where $V_i = \left[-\frac{1}{i}, \frac{1}{i}\right]$ for $i = 1,\ldots,4$.

1. First consider $x \in [-1,1]$. Then $x \in V_1$ so, by your definition of "union", $x \in V_1 \cup V_2 \cup V_3 \cup V_4$.

2. Next, consider $x \in V_1 \cup V_2 \cup V_3 \cup V_4$. Then $x$ must be in at least one of the $V_i$. No matter in which $V_i$ it is, note that always $x \ge -1$ and $x \le 1$, but this just means that $x \in [-1,1]$.

So, we have established $(\ast)$.

Note that we could also have considered the sets $W_i = \{V_i\}$ for $i = 1,\ldots,4$. That is, each $W_i$ is a set that has exactly one element, and this element is itself a set: $V_i \in W_i$ for $i = 1,\ldots,4$. Try to understand that in this case,
$W_1 \cup W_2 \cup W_3 \cup W_4 = \{V_1, V_2, V_3, V_4\},$
which is a set consisting of four elements, each element being a set by itself. In particular, the union of the $W_i$ is not equal to the interval $[-1,1]$.

In brief, all of those intervals, $$\displaystyle \left[-\frac{1}{2}, \frac{1}{2}\right]$$, $$\displaystyle \left[-\frac{1}{3},\frac{1}{3}\right]$$, etc. Are subsets of [-1, 1]. Every member of each interval is already in [-1, 1].

## 1. What is the difference between a union and a set in set theory?

A union in set theory combines two or more sets and creates a new set containing all the elements from each set. A set, on the other hand, is a collection of distinct objects. In a set, each element can only appear once, while in a union, elements can be repeated if they are present in multiple sets.

## 2. What are the basic rules for unions and sets?

The basic rules for unions and sets are:
1. Union of sets A and B, denoted as A ∪ B, is the set that contains all elements of A and B.
2. Intersection of sets A and B, denoted as A ∩ B, is the set that contains all elements that are common to both A and B.
3. Complement of set A, denoted as A', is the set that contains all elements that are not in A but are in the universal set.
4. Difference of sets A and B, denoted as A - B, is the set that contains all elements that are in A but not in B.

## 3. How are unions and sets used in real-life applications?

Unions and sets are used in various fields such as mathematics, computer science, and statistics. In mathematics, they are used to study the properties of numbers and other mathematical objects. In computer science, they are used to represent data and perform operations on it. In statistics, they are used to analyze data and make predictions.

## 4. What are some common mistakes when working with unions and sets?

Some common mistakes when working with unions and sets include:
1. Forgetting to include all elements in a union, resulting in a missing element in the final set.
2. Confusing intersection with union, which can lead to incorrect results.
3. Not understanding the concept of a universal set, which can result in errors when working with complements.
4. Using the wrong symbols or notation, which can cause confusion and errors in calculations.

There are various resources available for learning more about unions and sets, such as textbooks, online courses, and tutorials. Additionally, practicing with different examples and problems can help improve understanding. It is also helpful to ask questions and seek clarification from a teacher or tutor if needed.

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