Investigatory Project

Topics: Chemistry, Empty set, Basic concepts in set theory Pages: 7 (3205 words) Published: June 21, 2013
 
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|  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 square that represents the universal set U. Condition A U implies that there exists at least one element b U which is not in A. That element is somewhere outside of the drawn figure. Take a look to the all elements of set U which are not in A. They form a set which is called the complement of A regarding U and is denoted by AC . If we denote the set of all Blondies by U and by P the sentence “Mary belongs to the set of Blondies who do not speak French”, then the set of Blondies who do not speak French is a complement of the set of Blondies who speak French. So, AC is a set of those elements b in U which are not in A. It is denoted byAC = {b | b U, bA}     DISJOINT SETSIf sets A and B have no common elements i.e. no element of A is inB and no element of B is in A, then A and B are disjoint.In that case, set D is a set with no elements. The intersection ofsets A and B is the null set.Example: Suppose A and B are not comparable. If they are disjoint,they can be represented by the diagram on the left. If they are notdisjoint, they can be represented by the diagram on the right.   POWER SET AND PARTITION OF SETSometimes, for a better understanding of the characteristics of a set, it is useful to know some of its parts.Let A be a non-empty set. A power set of the set A is the set of all subsets of  A. A partition of the set A is a collection of non-empty disjoint subsets of A, whose union is A.Example for a partition of a set: Let X denote the set of all employees of one big company divided in several departments. We can say that employees are elements of disjoint subsets because each employee belongs to a different department. The union of those subsets is the set of all employees of the company. Each department has a chief. We can say that the chief is a representative of a subset.Let U be any non empty set and let P(U) denote the set of all subsets of U. Set P(U) is called the power set of the set U . Elements of the setP(U) are subsets of the set U.We say that...
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