Hey guys! What operation is this symbol?

**philleonard** · June 27th 2012, 07:59 AM

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;

Hey guys! What operation is this symbol?-2012-06-27-16.38.21.jpg

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

**Plato** · June 27th 2012, 08:14 AM

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;

Click image for larger version.

Name: 2012-06-27 16.38.21.jpg
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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.

**emakarov** · June 27th 2012, 08:25 AM

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, $\bigcup\{A_x: x\in X\}=\bigcup_{x\in X}A_x$ .

**richard1234** · June 27th 2012, 08:26 AM

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.

**philleonard** · June 27th 2012, 08:30 AM

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

**philleonard** · June 27th 2012, 08:34 AM

Thank you so much @emakarov

Fantastic explanation!

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