1
$\begingroup$

I'm struggling for the correct vocabulary/notation for the following situation. Let's say S is a (very simple) set with the following composition:

S = {1, 2, 3, 4, 5, 6}

Now, there are two subsets which compose this set, A and B:

A = {1, 2}
B = {3, 4, 5, 6}

Obviously, A is a subset of S, $A \subseteq S$, and B is a subset of S, $B \subseteq S$.

Not only are A and B subsets of S, these are subsets which uniquely compose S, i.e.

$A \cup B = S$ and $A \cap B = \varnothing$

(1) What is the vocabulary used to describe these type of subsets? A "unique" subset?

(2) If there is no special terminology, how would one best convey this concept notationally?

e.g.

$A \subseteq S$ and $B \subseteq S$ s.t. $A \cap B = \varnothing$

maybe be one possibility, but it doesn't show that the union of A and B are the set S...

Improve this question
$\endgroup$
  • $\begingroup$ A and B form a partition of S? $\endgroup$ – Air Conditioner Apr 8 '18 at 1:34
  • 1
    $\begingroup$ The word "partition" is what you want. An obvious caveat is that there will be many different possible partitions as one shouldn't think. Let $A, B$ be partions of $C$ mean that $A$ and $B$ have actually been specified. It hasn't be specified if A = {1,2,3} B ={4,5} or if A = {4} and B={1,2,3,5} or ... whatever. $\endgroup$ – fleablood Apr 8 '18 at 1:43
  • $\begingroup$ It's also worth being aware that equivalence relations define partitions of sets. en.wikipedia.org/wiki/Equivalence_relation $\endgroup$ – samerivertwice Apr 9 '18 at 17:53
  • $\begingroup$ You should probably add to your question $A\cup B=S$ which you intend but don't write. The requirements are $A\cup B=S$ and $A\cap B=\emptyset$. $\endgroup$ – samerivertwice Apr 9 '18 at 17:55
  • 1
    $\begingroup$ @ProducerofBS Edited $\endgroup$ – ShanZhengYang Apr 9 '18 at 18:43
5
$\begingroup$

I think you are looking for a partition:

A partition of a set X is a set of nonempty subsets of X such that every element x in X is in exactly one of these subsets...

Equivalently, a family of sets P is a partition of X if and only if all of the following conditions hold:

  • The family P does not contain the empty set
  • The union of the sets in P is equal to X ...
  • The intersection of any two distinct sets in P is empty

[Quoted from Wikipedia]

Improve this answer
$\endgroup$
3
$\begingroup$

The standard way to say this is that $\{A,B\}$ is a partition of $S$. In general, a partition of a set $S$ is a set of nonempty subsets of $S$ which are pairwise disjoint (the intersection of any two of them is empty) and whose union is $S$.

Another way to say this is that $A$ is a subset of $S$ and $B$ is the complement of $A$; that is, $B$ consists of exactly those elements of $S$ that are not in $A$. Some commonly used notations for the complement of $A$ (in $S$) are $S\setminus A$, $S-A$, or $A^c$. So your condition could be stated as $A\subseteq S$ and $B=S\setminus A$.

Improve this answer
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.