Mathematics Medicine Lukyanova Elena Anatolievna

Curriculum Introduction to Informatics Mathematics Introduction to theory of probability Informatics (Office) Theory of probability

Curriculum Math.Sets Math.Systems Informatics Informatics Math.Matrices Math.Sequences Informatics Math.Series Math.Control Math.Diff Informatics Math.Diff Math.Int Math.Int Informatics Informatics Math.Control Math.Probability Informatics Math.Probability Math.Probability Math.Probability Informatics Math.Control Informaticszachet Math.zachet

Instruction Doctor's smock (overall) NO PHONES NO FOOD NO DRINK

Symbolism of sets The set Union Is an element of Is not an element of The set of all x such that Intersection Is a subset of

Set notation A set is a collection of objects which are called its elements. If x is an element of the set S, we say that x belongs to S and write If y does not belong to S, we write The simplest way of specifying a set is by listing its elements. To denote the set whose elements are the real numbers use This notation is no use in specifying a set which has infinite number of elements. Such sets may be specified by naming the property which distinguishes elements of the set from objects which are not in the set. For example, the notation (Which should be read “the set of all x such that x>0” ) denotes the set of all positive real numbers.

Empty set, subset Empty set is set which has no elements. Notation for the empty set is . Note that this is not the same thing as writing, Let S and T are two sets, S is a subset of T if every element of S is also an element of T Notation for the subset is As an example, consider the sets which means that Q is an element of P. The elements of P are simply 1, 2, 3 and 4. But Q is not one of these. thenand

The Real Numbers Real numbers (R) - is set of numbers that can be named with decimals. The set of real numbers can be subclassified into two subsets: Rational Numbers (Q) and Irrational Numbers (Ir) Subset of Numbers SymbolDescription RationalQTerminating or nonterminating, repeating decimals IrrationalIrNonterminating, nonrepeating decimals The union of Ir and Q is the set of real numbers (R)

Rational Numbers Subset of NumbersSymbolDescription Natural NumbersN{1, 2, 3, 4, 5, …} Whole NumbersW{0, 1, 2, 3, 4, 5, …} IntegersZ{…, -2, -1, 0, 1, 2, …} Three subsets of Rational Numbers (Q) can be described by listing their elements The set of Whole Numbers and the set of Natural Numbers are subsets of Integers

The Real Numbers Z W 0 N QIr R 0,3333… 0,9  5, … This diagram illustrates how the sets N, W, Z, Q, Ir and R are related

Universal set, complement Universal set is the set which containing all the objects of interest in a particular context. Universal set is denote by  Given a set X and a universal set  we can define a new set, called the complement of X, denote The complement of X, contains all the elements that are not in X. Example Let  ={0, 1, 2, 3, 4, 5, 6, 7, 8, 9} and X={2, 4, 6, 8}. Then complement of X

Class exercises 1. List the elements of the following sets. A={all even numbers between 4 and 13 inclusive} B={all integers between 3,4 and 7,6 inclusive} 2. The sets X, Y and Z are defined by X={all positive odd numbers} Y={1, 2, 3, 4, 5} Z={all decimal digits} State whether the following are true or false 3. Given  ={0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, X={2, 4, 6, 8}, Y={1, 2, 3, 4, 5, 6, 7}. State

Home exercises 4. The sets , X, Y and Z are defined by  ={all positive integers} X={all positive even integers} Y={10, 20, 50} Z={5, 10, 15, 20} State whether the following are true or false 5. Given  ={2, 4, 6, 8, 10, 12}, X={2, 4, 6, 8}, Y={6, 8, 10}. State

Relationship between sets Intersection of X and Y, denoted The Intersection of X and Y contains the elements that are common to both sets Union of X and Y, denoted The Union of X and Y contains all the elements of both X and Y Example Let X={0, 1, 2, 3, 4, 5, 6, 7, 8, 9} and Y={0, 3, 6, 9, 12, 15}, then If X  Y= , that is, the intersection has no elements, then X and Y are disjoint sets

Class exercises 6. List the elements of the following finite sets. 7.Given A={5,6,7,9}, B={0,2,4,6,8} and  ={0,1,2,3,4,5,6,7,8,9} list the elements of each of the following sets A={x: x is a positive integer, greater than 5 and less than 11} B={x: x is odd and x is greater than 5 and less than 20} 8. Given a universal set, a non-empty set A and its complement, State which of the following are true and which are false A  =  A  =  A  =  A  = 

Home exercises 9. List the elements of the following finite sets. 10.Given A={5,6,7,9}, B={0,2,4,6,8} and  ={0,1,2,3,4,5,6,7,8,9} list the elements of each of the following sets C={x: x is between 10 and 50 inclusive and x is divisible by 3 and x is divisible by 5} 11. Given a universal set, a non-empty set A and its complement, State which of the following are true and which are false A  =A A  =A A  = 

Venn diagrams Sets can be represented diagrammatically using Venn diagrams. Union, intersection and complement can all be represented on Venn diagrams. The universal set, , usually represented by a rectangle; other sets are represented by circles.  A The universal set  and the set A  The complement of A A

Venn diagrams Intersection of A and B Union of A and B  XY X  Y= , X and Y are disjoint sets

Class exercises 12. A and B are intersecting sets. Represent on Venn diagram

Venn diagrams

Home exercises 13. A and B are disjoint sets. Represent the following on a Venn diagram