Defintion of The Union Of Sets

[wubie]

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.

 

[master_coda]

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

 

[wubie]

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.

 

[Warr]

u(AUB) = u(A) + u(B) - u(ANB) [The 'N' is intersect of A and B, in real life it looks like an upside-down 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.

 

[wubie]

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.

 

[himanshu121]

Originally posted by wubie
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?

u(AUB) represents the no. of elements in the set

 

[wubie]

Right! Thanks.

Now warr's post makes sense.

 

[himanshu121]

Originally posted by wubie
Right! Thanks.

Now warr's post makes sense.

It made sense before two, i was just giving him hints and i feel he has overlooked them

 

[nille40]

Perhaps this can be of some assistance:
$$A \cup B = \lbrace e \mid e \in A \lor e \in B \rbrace$$

The size of the union would be
$$|A \cup B| = |A| + |B| - |A \cap B|$$

Whch means "all the elements in $$A$$ + all the elements in $$B$$ - the elements in both $$A$$ and $$B$$".

Nille

 

[himanshu121]

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+BLUE-GREY 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

 

[phoenixthoth]

sumset

a more general union is this. let T be a collection of sets, usually at least two sets. for example, T={A,B}.

$$\bigcup T$$ is the set of elements in at least one member of T.

$$\bigcup \left\{ A,B\right\} =A\cup B$$ 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.

 

[wubie]

Thanks again everyone.

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