224 Generalised Union

Foundations of Mathematics

Math 224
Handout 10
§ 3.4
Collections, Generalised Union, and Generalised Intersection of Point-Sets
Dr. McLoughlin

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

We designate a new universe V, the power set of the original universe, as V = Ã( U ) or V = P( U ) V is the set of all subsets of U

For a generalised union or intersection of point-sets discussion we need to have U and point-sets defined, realise that V exists and is well defined based on U and we can go from there.

Let Y be a collection of point-sets which are all subsets of a well defined universe, U. Y Í P( U )

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

Let us consider some examples.

Example 1:

Let U = N , 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}. We will denote that a collection is a set whose elements are all point-sets. From whence does omega come?

Well, note W Í P( U ) = P( N )

So, for our above example we have ÇW = {1, 2} and ÈW = {1, 2, 3, 4, 5, 6, 7, 8} [so, in essence it is another way of writing the union or intersection of the point-sets A, B, and C - but it generalises - - - which is the point].

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

U = N , 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 a point-sets – they do not need to be elements of the universe, nor elements of any of the individual point-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} .

Here is yet another ditty:

Example 2:

Let U = N ₄ . So, U = {1, 2, 3, 4}. We designate a new universe V = P(N ₄ )

V = {Æ, {1}, {2}, {3}, {4}, {1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4}, {1, 2, 3}, {1, 2, 4}, {1, 3, 4}, {2, 3, 4}, {1, 2, 3, 4}}

and let us look at some Point-Sets: A = {1, 2} , B = {2, 3}, C = {3} and D = {1, 4}

Notice some interesting things: A Í U , A Ï U, but A Î V , and A Ú V; likewise for B, C, and D (e.g.: D Í U , D Ï U, but D Î V , and D Ú V)

Notice {A, B, C, D} Í V because they (the point-sets) are in (are members of) the power set V.

Notice 3 Ï V because 3 is a point so it is in a set but not a collection; notice 3 Î U because 3 is a point so it is in a set (the beginning universe is the 'basic' or 'fundamental' set).

Let L = {A, B, C, D}. So, ÇL = Æ and ÈL = {1, 2, 3, 4}

Nifty, eh?

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

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