Just as numbers can be added, subtracted, and multiplied, we can manipulate sets in certain
basic ways. The natural operations on sets are to combine their elements, to find those
elements common to both sets, and to determine which elements belong to one set but not another.
Just as graph theory is the mathematical study of graphs and their properties,
set theory is the mathematical study of sets and their properties.
Problem
If A and B are sets, then their union A∪B is the set comprising
any elements in either A or B; their intersection A∩B is the set of elements
in both A and B; and their set difference A−B is the set of elements in A but not in B.
Furthermore, if A is a subset of another set U, then the set complement of A with respect to U
is defined as the set Ac=U−A. See the Sample sections below for examples.
Given: A positive integer n (n≤20,000) and two subsets A and B of {1,2,…,n}.
Return: Six sets: A∪B, A∩B, A−B, B−A, Ac, and Bc
(where set complements are taken with respect to {1,2,…,n}).
Sample Dataset
10
{1, 2, 3, 4, 5}
{2, 8, 5, 10}
Sample Output
{1, 2, 3, 4, 5, 8, 10}
{2, 5}
{1, 3, 4}
{8, 10}
{8, 9, 10, 6, 7}
{1, 3, 4, 6, 7, 9}
From the definitions above, one can see that A∪B=B∪A and
A∩B=B∩A for all sets A and B, but it is not necessarily the case
that A−B=B−A (as seen in the Sample sections above).
This set theoretical fact parallels the arithmetical fact that addition is commutative
but subtraction is not.