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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;

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I know the union of sets, $\displaystyle A \cup B$ joins all elements in A and B. As far as I can tell from the description it is the union of all the sets $\displaystyle $A_{x}$$ where $\displaystyle x$ is an element of $\displaystyle X$. 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 $\displaystyle b \in C$ under the big union symbol, is this just a condition for the union operation?

Thanks guys :)

Phil

Re: Hey guys! What operation is this symbol?

Quote:

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;

Attachment 24174
I know the union of sets, $\displaystyle A \cup B$ joins all elements in A and B. As far as I can tell from the description it is the union of all the sets $\displaystyle $A_{x}$$ where $\displaystyle x$ is an element of $\displaystyle X$. 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 $\displaystyle b \in C$ 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 $\displaystyle |A\cup B\cup C|$ stands for the number of elements in at least one of $\displaystyle A,~B,\text{ or }C$.

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

Now please tell us what you need explaining.

Re: Hey guys! What operation is this symbol?

Quote:

Originally Posted by

**philleonard** As far as I can tell from the description it is the union of all the sets $\displaystyle $A_{x}$$ where $\displaystyle x$ is an element of $\displaystyle X$.

Yes, $\displaystyle \bigcup\{A_x:x\in X\}$ is the union of $\displaystyle A_x$ for all $\displaystyle x\in X$.

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Originally Posted by

**philleonard** could you then have a big intersect symbol to intersect the sets produced.

Yes, $\displaystyle \bigcup$ can be replaced by $\displaystyle \bigcap$ to mean the intersection of all $\displaystyle A_x$.

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Originally Posted by

**philleonard** There is also an example later with $\displaystyle b \in C$ under the big union symbol, is this just a condition for the union operation?

The big union symbol $\displaystyle \bigcup$ is used in two ways. One can write $\displaystyle \bigcup A$ if A is a family of sets: $\displaystyle A=\{A_1,\dots,A_n\}$. In this case, $\displaystyle \bigcup A$ means $\displaystyle A_1\cup\dots\cup A_n$. Alternatively, by analogy with the summation symbol $\displaystyle \sum$, one can write $\displaystyle \bigcup_{i\in I}A_i$ to mean the union of $\displaystyle A_i$ for all $\displaystyle i\in I$. Thus, $\displaystyle \bigcup\{A_x: x\in X\}=\bigcup_{x\in X}A_x$.

Re: Hey guys! What operation is this symbol?

Quote:

Originally Posted by

**philleonard** As far as I can tell from the description it is the union of all the sets $\displaystyle $A_{x}$$ where $\displaystyle x$ is an element of $\displaystyle X$. The lines being the cardinality of this set which has been joined (union).

That is correct. The |...| represent the magnitude of that set.

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 :)

Re: Hey guys! What operation is this symbol?

Thank you so much @**emakarov** :) Fantastic explanation!