MTH 426 Introduction to Measure Theory By Dr

MTH 426 Introduction to Measure Theory By Dr. Saqib Hussain 1

MTH 426 Lecture # 32 Review 2

Set A set is any well-defined list, collection, or class of objects. Examples: 3

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Venn-Euler Diagrams A simple and instructive way of illustrating the relationships between sets is in the use of so-called Venn-Euler diagrams. Here we represent a set by a plane area, 5

Line Diagrams Another useful and instructive way of illustrating the relationships between sets is in the use of so-called line diagrams. If A is a subset of B then we write B on a higher level. 6

Function Suppose that to each element in a set A there is assigned, by some manner or the other a unique element of a set B. We call such assignment a function. If we let f denote these assignments, then we write f: A->B 7

Relations A relation R consists of the following • a set A • a set B • An open sentence P(x, y) in which P(a, b) is true or false. We call R a relation from A to B and denote it by R=(A, B, P(a, b)) Notations: If P(a, b) is true we write a. Rb and if P(a, b) is false then we write a. Rb 8

Equivalence Relation A relation R in A is called an equivalence relation if a) R is reflexive b) R is symmetric c) R is transitive 9

Equivalent Sets Set A is equivalent to set B if there exists a function f: A->B which is both one-one and onto. Example: Let A={1, 2, 3} and B={a, b, c}. Then A B. 10

Infinite Set A set is said to be an infinite set if it is equivalent to a proper subset of itself. Example: The set of real number is an infinite set. 11