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Defintion of The Union Of Sets 
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#1
Jan1304, 09:54 PM

P: n/a

Hello,
I am having trouble interpreting the definition of the union of two sets as given in Modern Abstract Algebra in Schaum's Outlines. I can see by example but I can't seem to interpret the definition. Could someone reword this for me or give me another spin on this definition? Thankyou. Defintion as in Modern Abstract Algebra in Schaum's Outlines Let A and B be given sets. The set of all elements which belong to A alone or to B alone or to both A and B is called the union of A and B. I understand the example: Let A = {1,2,3,4} and B = {2,3,5,8,10}; then A union B = {1,2,3,4,5,8,10} And the way I interpret the union of two sets is this: Given two sets A and B, let the union of A and B be C. Then C contains the following: Elements common to both A and B. Elements in A and not in B. And elements in B but not in A. But I don't get the definition as given by Schaums. Any help is appreciated. Thanks again. 


#2
Jan1304, 11:01 PM

P: 678

I don't really understand what the difference is between your definition and Schaums. You're both saying the same thing.



#3
Jan1304, 11:36 PM

P: n/a

I don't know. Perhaps I am misinterpreting Schuam's definition. I don't see the equivalence between Schaum's definition and mine.
I interpret Schuams definition as follows: The set of all elements which belong to A alone. == The set consisting of just the elements of A. Meaning if the set consists of just the elements of A then B is an empty set. A union B = A The set of all elements which belong to B alone == The set consisting of just the elements of B. Similarly if the set consists of just the elements of B then A must be an empty set. B union A = B The set of all elements which belong to both A and B == The set consisting of the elements in both A and in B. This last part I can see. 


#4
Jan1304, 11:51 PM

P: 122

Defintion of The Union Of Sets
u(AUB) = u(A) + u(B)  u(ANB) [The 'N' is intersect of A and B, in real life it looks like an upsidedown U.]
The two different cases are whether A and B are discrete sets or not. If A and B do not have anything in commmon, there is no intersection, hence u(ANB) = {} = 0, the empty set. 


#5
Jan1404, 12:11 AM

P: n/a

Could you elaborate on your notation? I don't think I have seen the notation
u(AUB) before. Just what does the u(...) stand for? I understand that if A and B have nothing in common that their intersection is the empty set. I don't see the connection though. 


#7
Jan1404, 02:21 AM

P: n/a

Right! Thanks.
Now warr's post makes sense. 


#8
Jan1404, 02:42 AM

P: 658




#9
Jan1404, 06:54 AM

P: 34

Perhaps this can be of some assistance:
[tex]A \cup B = \lbrace e \mid e \in A \lor e \in B \rbrace[/tex] The size of the union would be [tex]A \cup B = A + B  A \cap B[/tex] Whch means "all the elements in [tex]A[/tex] + all the elements in [tex]B[/tex]  the elements in both [tex]A[/tex] and [tex]B[/tex]". Nille 


#10
Jan1404, 07:07 AM

P: 658

You can easily see from attachment that Union means The no. of ekements in both sets without repeating the same no in both sets
u can easily see Union will be RED+BLUEGREY this is because as i say not to repeat the common elements we subtract the Grey portion once coz when we add RED+BLUE they common elements are added up twice so we have to delete one 


#11
Jan1404, 04:14 PM

P: 1,572

a more general union is this. let T be a collection of sets, usually at least two sets. for example, T={A,B}.
[tex]\bigcup T[/tex] is the set of elements in at least one member of T. [tex]\bigcup \left\{ A,B\right\} =A\cup B[/tex] is the set of elements in at least one of A and B. ie, if x is in A or B (or both), then x is in the union. if T had three sets in it, the same definition would apply: x is in the union of three sets if it is in at least one of the sets in T. 


#12
Jan1504, 11:21 AM

P: n/a

Thanks again everyone.
These many different perspectives has given me a better understanding of the def. of union of sets. 


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