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view source for power set wikipedia jump to content main menu main menu move to sidebar hide navigation main page contents current events random article about wikipedia contact us contribute help learn to edit community portal recent changes upload file special pages search search appearance donate create account log in personal tools donate create account log in view source for power set add languages article talk english read edit view history tools tools move to sidebar hide actions read edit view history general what links here related changes upload file page information get shortened url switch to legacy parser in other projects wikidata item appearance move to sidebar hide power set you do not have permission to edit this page for the following reason this ip address range has been globally blocked this does not affect your ability to read wikipedia pages most people who see this message have done nothing wrong some kinds of blocks restrict editing from specific service providers or telecom companies in response to recent abuse or vandalism and can sometimes affect other users who are unrelated to that abuse review the information below for assistance if you do not believe that you have done anything wrong this block affects editing on all wikimedia wikis the ip address or range 5 135 0 0 16 has been globally blocked by jon kolbert for the following reason s open proxy webhost visit the faq if you are affected this block will expire on 04 29 8 january 2027 your current ip address is 5 135 42 194 even while globally blocked you will usually still be able to edit pages on meta wiki if you believe you were blocked by mistake you can find additional information and instructions in the stewards block wizard other useful links global blocks help i have been blocked you can view and copy the source of this page properties if math s is a finite set with the cardinality math 1 abs s n i e the number of all elements in the set math 1 s is math 1 n then the number of all the subsets of math s is math 1 abs itco mathcal p s 2 sup n sup this fact as well as the reason for the notation math 2 sup s sup denoting the power set math itco mathcal p s are demonstrated below an indicator function or a characteristic function of a subset math a of a set math s with the cardinality math 1 abs s n is a function from math s to the two element set math mset 0 1 denoted as math i sub a sub s mset 0 1 and it indicates whether an element of math s belongs to math a or not if math x in math s belongs to math a then math 1 i sub a sub x 1 and math 0 otherwise each subset math a of math s is identified by or equivalent to the indicator function math i sub a sub and math mset 0 1 sup s sup as the set of all the functions from math s to math mset 0 1 consists of all the indicator functions of all the subsets of math s in other words math mset 0 1 sup s sup is equivalent or bijection bijective to the power set math itco mathcal p s since each element in math s corresponds to either math 0 or math 1 under any function in math mset 0 1 sup s sup the number of all the functions in math mset 0 1 sup s sup is math 2 sup n sup since the number math 2 can be defined as math mset 0 1 see for example von neumann ordinals the math itco mathcal p s is also denoted as math 2 sup s sup obviously math 1 abs 2 sup s sup 2 sup abs s sup holds generally speaking math x sup y sup is the set of all functions from math y to math x and math 1 abs x sup y sup abs x sup abs y sup cantor s diagonal argument general sets cantor s diagonal argument shows that the power set of a set whether infinite or not always has strictly higher cardinality than the set itself or informally the power set must be larger than the original set in particular cantor s theorem shows that the power set of a countable set countably infinite set is uncountable uncountably infinite the power set of the set of natural number s can be put in a bijection one to one correspondence with the set of real number s see cardinality of the continuum the power set of a set math s together with the operations of union set theory union intersection set theory intersection and complement set theory complement is a σ algebra over math s and can be viewed as the prototypical example of a boolean algebra structure boolean algebra in fact one can show that any finite boolean algebra is isomorphic to the boolean algebra of the power set of a finite set for infinite boolean algebras this is no longer true but every infinite boolean algebra can be represented as a subalgebra of a power set boolean algebra see stone s representation theorem the power set of a set math s forms an abelian group when it is considered with the operation of symmetric difference with the empty set as the identity element and each set being its own inverse and a commutative monoid when considered with the operation of intersection with the entire set math s as the identity element it can hence be shown by proving the distributive property distributive laws that the power set considered together with both of these operations forms a boolean ring return to power set retrieved from https en wikipedia org wiki power_set privacy policy about wikipedia disclaimers contact wikipedia legal safety contacts code of conduct developers statistics cookie statement mobile view search search view source for power set add languages add topic