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

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 $\{1, 2, 3\}$?

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.

There is no difference; they are the same set.

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.

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

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

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

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

• 0 is certainly not the appropriate symbol – The Chaz 2.0 Dec 10 '14 at 0:59
• Principles of Mathematical Analysis by Rudin says "0" can represent an empty set – Jamil Santos Dec 10 '14 at 1:02
• Sorry for changing the 0 to $\emptyset$. I think $\emptyset$ is better thought, but it is your answer. – Thomas Dec 10 '14 at 1:06
• 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. – H_B Dec 10 '14 at 1:08
• 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 – Jamil Santos Dec 10 '14 at 1:08