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B.Sc. Second Year, Paper – VI ( Real Analysis –I )

“ Gramin ACS College Vasantnagar, Kotgyal”

Dr. Parshuram R. Shnde

Department of Mathematics

Gramin ACS College Vasantnagar, Kotgyal.

Tq : Mukhed, Dist : Nanded (M.S.) INDIA

Unit – I : Sets and Properties PPT Presented by……

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Unit – I : Sets and Properties

Contents :

Field structure and order structure, Intervals. Bounded and unbounded sets, Supremum, Infimum. Completeness in the set of real numbers, Order completeness in R, Archimedean property of real numbers, Dedekind's Property, Complete-ordered field, Representation of real numbers as points of a straight line. Neighbourhood of a point, Interior point of a set, Open set Limit point of a set, Bolzano-Weierstrass theorem. Closed sets, Closure of a set, Dense sets, Some important theorems. Countable and uncountable sets.

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Field Structure and Order Structure :

Field Structure :

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Order Structure :

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Intervals – Open and Closed

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Bounded And Unbounded Sets : Supremum, Infimum

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Completeness In The Set of Real Numbers

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Theorem : The set of rational numbers is not order-complete

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Archimedean Property of Real Numbers

Theorem : The real number field is Archimedean, i.e., if a and b are any two positive real numbers then there exists a positive integer n such that na > b.

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Theorem : Every open interval ]a,b[ contains a rational number.

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Dedekind’s Form of Completeness Property

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Explicit Statement of the Properties of the Set of Real Numbers as a Complete-OrderField

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Representation of a Real Numbers as Points on a Straight Line

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Neighbourhood of a Point

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Interior Points of a Set

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Open Set

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Example : Show that every open interval is an open set. Or every open interval is a nbd of each of its points

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Example : Show that every open set is union of open intervals

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Theorem : The interior of a set is an open set.

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Theorem : The interior of a set S is the largest open subset of S.

OR

The interior of a set S contains every open subset of S.

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Theorem : The union of an arbitrary family of open sets is open

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Theorem : The Intersection of any finite number of open sets is open.

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Limit Points of a Set

Bolzano-Weierstrass Theorem :

“Every infinite bounded set has a limit Point” ( Statement Only)

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Closed Set :-

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Dense Set :-

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Some Important Theorems :-

Theorem : A Set is closed iff its complement is open

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Theorem : The Intersection of an arbitrary family of closed sets is closed

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Theorem : The union of two closed sets is a closed set.

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Theorem : The derived set of a set is closed.

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Theorem : A closed set either contains an interval orelse is nowhere dense.

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Countable and uncountable sets :-

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Theorem : The set of real numbers in [0,1] is uncountable

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Theorem : The set of rational numbers in [0,1] is countable

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Theorem : If ** f : A→B is one-to-one one B is countable then A is countable

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Thank you…