# Thread: Hey guys! What operation is this symbol?

1. ## Hey guys! What operation is this symbol?

Hey guys. To introduce myself to the forum, I'm a first (going on second) year Computer Science student from Liverpool UK. Just a quick question on a symbol I have come across;

I know the union of sets, $A \cup B$ joins all elements in A and B. As far as I can tell from the description it is the union of all the sets $A_{x}$ where $x$ is an element of $X$. The lines being the cardinality of this set which has been joined (union).
Am I close at all to it's meaning? If so, could you then have a big intersect symbol to intersect the sets produced. There is also an example later with $b \in C$ under the big union symbol, is this just a condition for the union operation?

Thanks guys

Phil

2. ## Re: Hey guys! What operation is this symbol?

Originally Posted by philleonard
Hey guys. To introduce myself to the forum, I'm a first (going on second) year Computer Science student from Liverpool UK. Just a quick question on a symbol I have come across;

I know the union of sets, $A \cup B$ joins all elements in A and B. As far as I can tell from the description it is the union of all the sets $A_{x}$ where $x$ is an element of $X$. The lines being the cardinality of this set which has been joined (union).
Am I close at all to it's meaning? If so, could you then have a big intersect symbol to intersect the sets produced. There is also an example later with $b \in C$ under the big union symbol, is this just a condition for the union operation?
You realize that it is very hard to read the image.

The notation $|A\cup B\cup C|$ stands for the number of elements in at least one of $A,~B,\text{ or }C$.

It equal to $|A|+|B|+| C|-|A\cap B|-|A\cap C|-| B\cap C|+|A\cap B\cap C|$.

Now please tell us what you need explaining.

3. ## Re: Hey guys! What operation is this symbol?

Originally Posted by philleonard
As far as I can tell from the description it is the union of all the sets $A_{x}$ where $x$ is an element of $X$.
Yes, $\bigcup\{A_x:x\in X\}$ is the union of $A_x$ for all $x\in X$.

Originally Posted by philleonard
could you then have a big intersect symbol to intersect the sets produced.
Yes, $\bigcup$ can be replaced by $\bigcap$ to mean the intersection of all $A_x$.

Originally Posted by philleonard
There is also an example later with $b \in C$ under the big union symbol, is this just a condition for the union operation?
The big union symbol $\bigcup$ is used in two ways. One can write $\bigcup A$ if A is a family of sets: $A=\{A_1,\dots,A_n\}$. In this case, $\bigcup A$ means $A_1\cup\dots\cup A_n$. Alternatively, by analogy with the summation symbol $\sum$, one can write $\bigcup_{i\in I}A_i$ to mean the union of $A_i$ for all $i\in I$. Thus, $\bigcup\{A_x: x\in X\}=\bigcup_{x\in X}A_x$.

4. ## Re: Hey guys! What operation is this symbol?

Originally Posted by philleonard
As far as I can tell from the description it is the union of all the sets $A_{x}$ where $x$ is an element of $X$. The lines being the cardinality of this set which has been joined (union).
That is correct. The |...| represent the magnitude of that set.

5. ## Re: Hey guys! What operation is this symbol?

Click on the image for a larger view please. Thanks for taking time to explain that though.
Edit: Sorry just seen all the new posts too, will read now. Thanks

6. ## Re: Hey guys! What operation is this symbol?

Thank you so much @emakarov Fantastic explanation!