255 Generalised Union

Set Theory Math 255
Handout 10
§ 3.4
Collections, Generalised Union, and Generalised Intersection of Sets
Dr. McLoughlin

Let U designate a well defined universe and A, B, and C sets within the universe.

For a generalised union or intersection of sets discussion let us consider some examples first.

Let U = IN , and A = {1, 2, 3, 4, 5, 6, 7} , B = {1, 2, 3, 5, 7}, and C = {1, 2, 4, 6, 8} .

Clearly A È B È C = {1, 2, 3, 4, 5, 6, 7, 8}

and A Ç B Ç C = {1, 2}

Let us define the collection W = {A, B, C}. Note that a collection is a set whose elements are all sets. From whence does omega come? Well, note W Í Ã( U ) = Ã( IN )

Let Y be a collection of sets which are all subsets of a well defined universe, U.

Define ÇY = { x | x Î K for all sets K Î Y }. Define ÈY = { y | y Î M for some set M Î Y }.

So, for our above example we have ÇW = {1, 2} and ÈW = {1, 2, 3, 4, 5, 6, 7, 8}

Now, sometimes we have a third set defined, which is called an index set. So, suppose

U = IN , and A₁ = {1, 2, 3, 4, 5, 6, 7} , A₂ = {1, 2, 3, 5, 7}, and A₃ = {1, 2, 4, 6, 8} . We say that the index set

is the set consisting of {1, 2, 3} [NOTE: these elements are subscripts of the sets – they do not need to be elements of the universe, nor elements of any of the individual sets!!!]

We can write the generalised union and intersection, respectively, without the use of the notation for a collection by saying

= {1, 2, 3, 4, 5, 6, 7, 8} and = {1, 2} . Nonetheless, we could have also denoted the index set. Let us denote the index set as D. So, D = {1, 2, 3}. If we reference this index set then the generalised union and intersection, respectively, as = {1, 2, 3, 4, 5, 6, 7, 8} and = {1, 2} .

Page 100: S14 and S15 are wrong! Please correct them.

Links:

Back to page 1

Back to Math 255 main page