ABSTRACT
If A is a proper subset of a set B we write A ⊂ B. If equality is possible we write A ⊆ B. If A _ _ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003071723/fddffade-8341-4eba-9b46-492985c90eb0/content/eq1.tif"/> is a family of subsets of a given set A then ∩ A _ _ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003071723/fddffade-8341-4eba-9b46-492985c90eb0/content/eq2.tif"/> and ∪ A _ _ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003071723/fddffade-8341-4eba-9b46-492985c90eb0/content/eq3.tif"/> will respectively denote the intersection and union of all members of A _ _ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003071723/fddffade-8341-4eba-9b46-492985c90eb0/content/eq4.tif"/> . In the case that A _ _ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003071723/fddffade-8341-4eba-9b46-492985c90eb0/content/eq5.tif"/> is empty then ∩ A _ _ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003071723/fddffade-8341-4eba-9b46-492985c90eb0/content/eq6.tif"/> = A and ∪ A _ _ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003071723/fddffade-8341-4eba-9b46-492985c90eb0/content/eq7.tif"/> = ø. If A is a set we denote the family of all subsets of A by ℙ (A). If f is a function from a set A to a set B then f induces a function ℙ (f) from ℙ (A) to ℙ (B) defined as follows: if A′ ϵ ℙ (A) then ℙ (f)(A′) = {f(a′) | a′ ϵ A′}. Therefore ℙ (_) can be considered as an endofunctor of the category of sets.
