Pirates & Revolutionaries: Wikibooks: Set Theory, “Some properties of set operations”, and “Families of sets”, summary

by Corry Shores

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Summary of

Wikibooks: Set Theory

Sets

Some properties of set operations

and

Families of sets

Brief summary:
Union and intersection are each symmetrical (so we can invert the sides the terms are on) and associative (so we can change the parenthesized groupings). Also, union distributes over intersection and intersection distributes over union. De Morgan’s law says that a set’s relative complement to a united pair is the same as the intersection of the first set’s relative complement to one pair member with that of the other. Also, a set’s relative complement to some intersected pair is the same as the union of the first set’s relative complement to one pair member with that of the other. A set of sets is called a family or a collection, and we can find the union and intersection of their member sets.

Summary

Some properties of set operations

Now having established certain concepts regarding sets [union, inclusion, complement, etc., see the prior section] we can discuss certain properties of sets and operations that we may conduct on them.

Union and intersection

The first property is that union and intersection are symmetric [so we can switch the side where we list the set] and they are associative [so we can change where we put the parentheses and thus the organization of their grouping.]

The union and intersection operations are symmetric. That is, if A and B are sets,

$A \cap B = B \cap A$

$A \cup B = B \cup A$

Furthermore, they are associative. That is, if A, B, and C are sets,

$(A \cap B) \cap C = A \cap (B \cap C)$

$(A \cup B) \cup C = A \cup (B \cup C)$

[Wikibooks]

[Also, when we have a union of a set with two other intersecting sets, or if we have the intersection of a set with two other uniting sets, then we can distribute the first term over the two in the other pairing.]

Furthermore, union distributes over intersection and intersection distributes over union. That is, if A, B, and C are sets,

$A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$

$A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$
[Wikibooks]

De Morgan’s laws

[Let me first quote this part on De Morgan’s laws:]

Two important propositions for sets are De Morgan's laws. They state that, for sets A, B, and C,

$A \setminus (B \cup C) = (A \setminus B) \cap (A \setminus C)$

$A \setminus (B \cap C) = (A \setminus B) \cup (A \setminus C)$
[Wikibooks]

[So consider if set A is {1, 2, 3, 4, 5, 6}, set B is {1, 2} and set C is {5, 6}. First let us unite B and C, that is B∪C, to get {1, 2, 5, 6}. Then we want to know, what is in A that is not in B∪C, that is: A\(B∪C) ? That would be {3, 4}. Let us instead ask, what is in A but not in B, that is: A\B ? This would be {3, 4, 5, 6}. And what is in A but not in C, or A\C? That would be {1, 2, 3, 4}. Now, what happens when we intersect these two sets, or (A\B) ∩ (A\C) ? Again, we get {3, 4}, because these are the terms that both share in common. Similar reasoning can illustrate how the second formula holds as well.] [Now, recall the notion of absolute complement. If instead of some A we are interested in everything in the universe which is not in some intersection or union of sets, then the same laws can be expressed this way:]

When A is a universe to which B and C belong, De Morgan's laws can be stated more simply as,

$(B \cup C)^C = B^C \cap C^C$

$(B \cap C)^C = B^C \cup C^C$
[Wikibooks]

Families of sets

[I am not certain I grasp this next concept. It seems to be the following. We can have sets of other sets. In this case, for some reason there is something special about this grouping of sets. It seems more than merely how each set is already constituted by every possible subset grouping of its elements. Perhaps the idea is that each of the sets in the larger grouping has a certain character and importance. For example, if we had a group including the set of natural numbers, and rational numbers, and other very important sets of numbers. But I am just guessing. At any rate, the larger grouping is called a family or collection of sets. What the following seems to be doing is showing us how we notate the union of all sets within a family, or how we notate the intersection of all the included sets. But I am not sure. It might be saying something else, like the union of families for example. See the quotation below.]

A set of sets is usually referred to as a family or collection of sets. Often, families of sets are written with either a script or Fraktur font to easily distinguish them from other sets. For a family of sets $\mathfrak{A}$ , define the union and intersection of the family by,

$\bigcup \mathfrak{A} = \bigcup_{A \in \mathfrak{A}} A = \{x | x \mbox{ is in some } A \in \mathfrak{A}\}$

$\bigcap \mathfrak{A} = \bigcap_{A \in \mathfrak{A}} A = \{x | x \mbox{ is in all } A \in \mathfrak{A}\}$

For a family of sets, we say that it is pairwise disjoint if any two distinct sets we choose from the family are disjoint.
[Wikibooks]

Source (of text and images):

Wikibooks. “Set Theory/Sets.” <https://en.wikibooks.org/wiki/Set_Theory>
(Accessed 25-03-2016)

My Academia.edu Page w/ Publications

25 Mar 2016

Wikibooks: Set Theory, “Some properties of set operations”, and “Families of sets”, summary

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