Investigatory Project

| | | ALGEBRA OF SETSBASIC OPERATIONSAs we have introduced meaning of the terms set, subset, null set and universal set, we can learn how to build new sets using the sets we already know. The way we do it is called set operations.The set operations are: union, intersection, difference and complement. They are called Boolean operations.Let A and B be subsets of the universal set U. Then we can look at the set C U that contains all elements of set A and all elements of set B and nothing else. In other words, x C implies two statements: x A or xB. Shortly, we describe set C withC = { x | xU, x A or xB}UnionThe union of sets A and B is the set of all elements which belong to A, to B or to both. It is denoted by A B,AB={x | x U, xA or x B}.The conjunction “or” permits that x is an element in set A and an element in set B. We can say that x is an element of the set AB because of the three reasons: if xA, then xA B; if xB, then x AB; if xA and x B, then xA B.In the other words, we can observe the set C U, which contains all elements of set A and all elements of set B, but no other elements. C = AB.The union of sets is the smallest subset of U, which embraces sets A and B as its subsets.IntersectionHowever, sets A and B also define a set of elements that are common to both of them.The intersection of sets A and B is the set of elements that are common to sets A and B. It is denoted by A B and is also a subset of U.AB ={x | x U, xA and x B}Thus, for sets A, B U completely defined is the set D, which consists of those and only those elements of U that are in A and in B at the same time.Set D is the intersection of sets A and B.D= A B. DifferenceThe difference of sets A and B is the set of elements which belong to A, but do not belong to B. It is denoted by A – B or A \ B.A \ B = {x | x U, x A and x B} Complement SetLet U be the universal set and AU, A U set whose elements are all points inside the figure that is a part of the