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I have a simple question related to set theory. Is there any difference between the following two sets?

What is the difference between $\{1, 2\}\cup\{3\}$ and $\{1, 2, 3\}$?

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Others have already pointed out that the sets are the same.

Remember that two sets are the same when they have the same elements. That is, two sets $X$ and $Y$ are the same exactly when $a$ is an elements of $X$ if and only if $a$ is an element of $Y$.

So consider the two sets $X = \{1,2\}\cup \{3\}$ and $Y= \{1,2,3\}$. The elements of $X$ are exactly $1, 2,$ and $3$. The elements of $Y$ are exactly the same. So the two sets are equal.

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There is no difference; they are the same set.

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The way that we show two sets are equal is by showing that every thing in one set is in the other set, and vice versa.

i.e. $$A=B \text{ if and only if } A\subseteq B \text{ and }B\subseteq A$$

Since union is defined to be the set of elements in both sets in the union, it is obvious in your situation that the two sets in question are equal.

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The union operator $\cup$ joins the sets $\{1, 2\}$ and $\{3\}$ to form the set $\{1, 2, 3\}$. So

$$ \{1, 2\} \cup \{3\} \equiv \{1, 2, 3\}. $$

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Since the sets are equal, the difference (in either direction) is equal to $\varnothing$.

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The union of $\{1,2\}$ and $\{3\}$ is equal to $\{1,2,3\}$. The intersect of $\{1,2\}$ and $\{3\}$ would be equal to $0$.

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    $\begingroup$ 0 is certainly not the appropriate symbol $\endgroup$ – The Chaz 2.0 Dec 10 '14 at 0:59
  • $\begingroup$ Principles of Mathematical Analysis by Rudin says "0" can represent an empty set $\endgroup$ – Jamil Santos Dec 10 '14 at 1:02
  • $\begingroup$ Sorry for changing the 0 to $\emptyset$. I think $\emptyset$ is better thought, but it is your answer. $\endgroup$ – Thomas Dec 10 '14 at 1:06
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    $\begingroup$ Yes, but when the user is trying to figure out something as elementary as this, it would probably be best to not confuse them; many people have a hard time understanding the difference between $\{\}$ and $\{0\}$ at first. $\endgroup$ – H_B Dec 10 '14 at 1:08
  • $\begingroup$ No problem. I'm open to feedback. I just saw the zero in the rudin book. If the consensus is the empty set symbol is more appropriate, I'm fine with my answer being changed $\endgroup$ – Jamil Santos Dec 10 '14 at 1:08

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