MHB Union & Set Rules - Learn the Basics Now!

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The discussion clarifies the concept of unions in set theory, specifically addressing confusion about how the union of multiple sets can equal just one of those sets. The union combines all unique elements from the involved sets, meaning overlapping elements are only counted once. For example, the union of intervals like [-1,1] and [-1/2,1/2] results in [-1,1] because all elements of the latter are already included in the former. The explanation emphasizes that while individual sets can contain overlapping elements, their union encompasses the entirety of the range covered by those sets. Understanding this fundamental principle resolves the initial confusion about unions and intersections.
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I am going over some of my notes, and I cannot understand unions, here is the selection I am having trouble with
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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.
Thanks for your help!
 

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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.
Thanks for your help!

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, [math]\left[-\frac{1}{2}, \frac{1}{2}\right][/math], [math]\left[-\frac{1}{3},\frac{1}{3}\right][/math], etc. Are subsets of [-1, 1]. Every member of each interval is already in [-1, 1].
 
I was reading documentation about the soundness and completeness of logic formal systems. Consider the following $$\vdash_S \phi$$ where ##S## is the proof-system making part the formal system and ##\phi## is a wff (well formed formula) of the formal language. Note the blank on left of the turnstile symbol ##\vdash_S##, as far as I can tell it actually represents the empty set. So what does it mean ? I guess it actually means ##\phi## is a theorem of the formal system, i.e. there is a...

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