7.Nepal Mathematics Centre(NMC-CAPS)

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Kamana Shrestha CAPS Welcome To

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FREE FREE Saturday Special (Saturday, November 5, 2011) (11.30 a.m. – 12.30 p.m.) Lecture On Basic Mathematics With Special Focus On Logic, Set Theory and Real Number System By Prof. Dr. R. M. Shreshtha CAPS

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12/19/2011 5 NAMASTE 2011 On Behalf of Centre for Academic and Profession al Services And Nepal Mathematics Centre is pleased to welcome you all at CAPS Lecture Series –(1-3)

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Practice makes perfect..But nobody's perfect......so why practice? Your future depends on your dreams,So go to sleep Inductive? ? Deductive? L O G I C

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Logical Connectives Thru A or B or Both Conjunction Thru A and C Disjunction A A B C Door Logic Function Logic using Doors C

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Logical using Switches On A and B On A or B or Both Conjunction Disjunction A and B Input Switches Switches Logic Function

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Truth-Table An Input-Output-Table For Conjunction A A and B are input switches; and C is the output LED Truth- Value C B A is OFF : F or 0: Truth- Table A B C 0 0 0 B is OFF : F or 0: C is OFF : F or 0 A is OFF (0) and :B is OFF (0): So, C is OFF (0).

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Truth-Table An Input-Output-Table For Conjunction A A and B are input switches; and C is the output LED Truth- Value C B A is OFF : F or 0: Truth- Table A B C 0 0 0 B is ON : T or 1: C is OFF : F or 0 A is OFF (0) and :B is ON (1): So, C is OFF (0). 0 1 0

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Truth-Table An Input-Output-Table For Conjunction A A and B are input switches; and C is the output LED Truth- Value C B A is ON : T or 1: Truth- Table A B C 0 0 0 B is OFF : T or 1: C is OFF : F or 0 A is ON (1) and :B is OFF (0): So, C is OFF (0). 0 1 0 1 0 0

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Truth-Table An Input-Output-Table For Conjunction A A and B are input switches; and C is the output LED Truth- Value C B A is ON : T or 1: Truth- Table A B C 0 0 0 B is ON : T or 1: C is ON : T or 1 A is ON (1) and :B is ON (1): So, C is ON (1). 0 1 0 1 0 0 1 1 1

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COLLECTION ENSEMBLE SET AGGREGATE GROUP FAMILY BUNCH DECK

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Sun Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune The Solar System { } Solar system Earth  Solar system Neptune  Solar system Basic Concepts Set and Set-membership Sun is in or belongs to Solar system  Sun Earth is in or belongs to Solar system Neptune is in or belongs to Solar system

Diagram1923 : 

Diagram1923 0 -1 1 2 3 4 5 6 7 8 9 Integers from -1 to 9 Positive integers less than 10 Even integers from 2 to 9 Odd integers from 1 to 9 Primes <10 J o h n V e n n Venn1834

The Solar System : 

Inner planets Outer planets Superior planets Sun The Solar System Mars {Sun} {Sun} is a subset of Solar System Solar System  {Inner Planets} is a subset of Solar System {Inner planets}  Solar System {Outer planets} is a subset of Solar System {Outer planets}  Solar System

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{Outer Planets} Mars JupiterSaturnUranusNeptune Set Union {Mars}  {Outer Planets} = {Superior Planets} {Mars}

{Inner Planets} : 

{Inner Planets} {Superior Planets} MercuryVenusEarthMars MarsJupiterSaturnUranusNeptune SETINTERSECT ION I  S = {Mars}

Subset : 

0 1 2 3 4 0 1 2 T S Subset

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2. a) 0 1 2 3 4 0 1 2 T T 0 1 2 2. b) 0 1 2 S S = T S

Pioneers of Set Theory : 

Pioneers of Set Theory

The Great Beginning : 

N U M B E R S AND N U M B E R – S Y M B O L S The Great Beginning

Ancient Artifacts : 

Ancient Artifacts

Babylonian and EgyptianNumerals and Numbers : 

Babylonian and EgyptianNumerals and Numbers Plimpton Tablet 322 Moscow Papyrus

Chinese and HarappanNumerals and Numbers : 

Chinese and HarappanNumerals and Numbers Oracular Bone Harappa Tablets

Some EarlyNumerals and Numbers : 

Some EarlyNumerals and Numbers

Development of Numbers : 

Development of Numbers Starting from Natural (or Counting) Numbers  N = {1, 2, 3, 4, 5, ...} , Whole Numbers  {0, 1, 2, 3, 4, 5, ...} Integers Z = {..., -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, ...} Irrational Numbers Q* = {x | x is a real number that is not rational} we arrive at Rational numbers Q = {a/b: a,b  Z, b ≠ 0},

Number line : 

Number line + ─ 0 1 -1 √2 -√2 e 3 π 2 1/2

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Suppose a and b are two given real numbers or points on the real number line with a < b. + ─ 0 1 -1 a 3 b 2

Finite Intervals : 

Finite Intervals

Infinite Intervals : 

Infinite Intervals

Where to Begin? and How to Begin? : 

Where to Begin? and How to Begin? Relations, Functions and Graphs ?

Graph Models : 

Graph Models A network of (bi-directional) road links connecting a set of cities (K, B), Kathmandu Biratnagar Nepal Pokhara (K, P), (K, S) Surkhet Ordered Pairs:

A Table : 

A Table 1 2 3 2 3 4 X X X X X X (1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4) { } Set of Ordered Pairs

Matrix : 

Matrix and A= {1, 2, 3} B= { 2, 3, 4} (1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 3), (2, 2), (3, 2), (3, 4) { } 0 1 1 1 0 1 1 1 0 1 1 1 0 0 0 0 M = R

Digraphs : 

Digraphs V = {a, b, c, d}, R = { }. (a, b), (a, d), (c, a), (b, b), (b, d), (c, b), (d, b) a b c d

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Ordered Pairs, Cartesian Product and Relation {(1, 2), (1, 3), (1, 4), (2, 3), 1 2 3 2 3 4 Relation: {(1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4)} Table: Domain D = {1, 2, 3}, Range R = {2, 3, 4} Sets : X = {1, 2, 3}, Y = {2, 3, 4} (2, 4), Arrow Diagram: R Cartesian Product: {(1, 2), (1, 3), (1, 4), (2, 2), (2, 3), (2, 4), (3, 2), (3, 3), (3, 4)} Ordered pairs: (1, 2), (2, 3), (3, 4), etc. (3, 4)} Graph: x y 0 R = { y = f (x): x < y and x, y S ={1, 2, 3, 4}

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Input(Fruit) Process(Squeezing) Output(Fruit juice)

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Input (Fruit) Function (Squeezing) (Squeezing) Output (Fruit Juice) Apple Banana Cherry . . Grape Apple Juice Banana Juice Cherry Juice . . Grape Juice Input (Fruit) Output (Fruit Juice) Function (Squeezing)

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Input A Bit More Mathematical (Add 7) Output Rule in Words (Function) Add 7 to the Value x Input x Output x + 7 Arrow Mapping/Diagram Table Graph Equation: y = x + 7

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R a n g e R a n g e x-axis y-axis 0 a b x  Domain [a, b] Function (f) Function   f : [a, b]  [, ]. y = f(x)  [, ]

Vertical line test: : 

Vertical line test: A graph in a coordinate plane represents a function if and only if it intersects the graph at only one point. x-axis y-axis Graph O    

Three points of intersection : 

Each of the first three vertical lines cuts the following graph at one and only one point, but the fourth one cuts it at three points. So the graph does not represent a function Three points of intersection x-axis y-axis O   

Representations of Functions : 

Representations of Functions x a e f (x) f (e) f (a) 1 -1 2 - 2 3 -3 Many-to-one (onto ) One-to-one 1 4 1 -1 2 -2 1. Arrow diagrams: One-to-one (into) f f f A B A B A B

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Function in a Nutshell Sets : A = {-3, -1, 1, 3, 4}, B = {3, 2, 1, -2} Relation: { (-3, 3), (-3, 2), (-1, 2), (1, 1), (3, -2), (4, -2)} Function as a special kind of relation: Function as a graph: (-3, 3), (-1, 2), (1, 1), (3, -2), (4, -2) { } Function as table: Domain {-3, -1, 1, 3, 4} Range {-2, 1, 2, 3} Arrow diagram or Mapping: -3 -1 1 3 4 3 2 1 -2 R

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Domain of f = Range of f -1 Range of f = Domain of f -1 Inverse function Domain and Range of inverse function This is illustrated in the following figure:

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Inverse function computation scheme Suppose f: A → B is a given function Let us now see how we can find The scheme below illustrates the procedure: f -1

Composition of functions : 

Composition of functions x ● gof = g(f) f g A B C f (x) ● ● g(f(x))

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y-axis 0 y = f(x) = 1 Unit-valued function x-axis y-axis 0 y = f(x) = x Graph of Identity Function x-axis y-axis 0 y = f(x) =  x  Absolute Value Function x-axis y-axis

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Graph of the greatest integer function. x-axis y-axis 0 y = f(x) = [x] Stair-case function

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O O - Graphs of Quadratic Functions  x y -  x y

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Three zeros of a cubic function R a n g e y  - Domain x - axis y - axis O -3 2 -1 x  - (0, -6) y-intercept -6 Domain, Range and Intercepts on the Axes

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12/19/2011 55 NAMASTE 2011 On Behalf of Centre for Academic and Profession al Services And Nepal Mathematics Centre is pleased to welcome you all at CAPS Lecture Series --3

Where to Begin? and How to Begin? : 

Where to Begin? and How to Begin? ? CURVE SKETCHING

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SOMETHING NON-MATHEMATICAL AND SOMETHING MATHEMATICAL We begin with

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All Time Experiences

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Rainbow

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The Milky Way or The spiral galaxy

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Early Interests in Curves Bhimbetka rock painting (India) Stone Age Painting The Fighting Cats, which is about 10,000 years old, is in serious danger of crumbling to pieces (LibyanSahara-Prehistoric Art) Paleolithic Bulls and Other Animals Crowd Calcite Walls (Lascaux, France) Three Headed Seal , Indus Civilization Harrapan Plate , Indus Civilization A plate from Nepal, the decoration show a saucer-like shape and a large headed humanoid.  (ca.3000 B.C.)

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Kalachakra Mandala Historic painting of Swayambhu Nath Temple and its vicinity (Artist unknown) Image by Daniel Wright Swayambhu Temple Today Most common cave and rock arts found in widely separated locations on different continents

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Slide 4 Intercepts on the Axes  

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Symmetry about an Axis

Is a butterfly symmetrical? : 

Is a butterfly symmetrical?

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O O -

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Three zeros of a cubic function R a n g e y  - Domain x - axis y - axis O -3 2 -1 x  - (0, -6) y-intercept -6 Domain, Range and Intercepts on the Axes

Poles and Holes : 

Poles and Holes f(a) → ∞ Hole at x = a Asymptote : x = a Pole at x = a f(a) undfined a (a, 2a) f(x) = 1/(x – a)

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x y O (0, 1) .7 2.7 - .7 -2.7 -2 2 -3 0 -  +  Symmetry f(x) = - f(-x)

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x y O (0, 1) x- intercepts: y = 0  x  .7 and  2.7 .7 2.7 - .7 -2.7 + + + + + + + + + + + + + + -2 2 -3 Slope > or = or < 0; Intervals of Increase/Decrease 0 y-intercept x = 0  y = 1 -  +  Slope = 0 at x  2, 0

Fascination of Curvesin Ancient Times : 

Fascination of Curvesin Ancient Times Man got fascinated with curves long before they became the subject of mathematical study. This can be seen in numerous examples of their decorative use in art and on everyday objects dating back to prehistoric times.[1] Curves, or at least their graphical representations, are simple to create, for example by a stick in the sand on a beach. Path of snake on a desert

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Curve Sketching Region of existence ( domain and range) Passage through certain points on the x-axis and y-axis ( x- and y- intercepts) Symmetry with respect to certain lines ( lines parallel to the axes, slanting line, and origin) and as decided by odd or even or periodic nature of the function Sketching a curve is just having a qualitatively good looking representation that will help visualize certain behaviour such as

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Curve Sketching 4 Closeness or nearness to certain line or lines but not meeting the line (Asymptote), Rise and fall, (increasing or decreasing nature of the function - monotonocity) Extremum value (maximum or minimum value, point where the slope of the tangent is zero and switches from increasing to decreasing or vice versa),

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Curve Sketching 7. Changes in concavity (Changes form CU to CD or vice versa – inflection) 8. Sketch (free-hand drawing or putting together some or all of the above information).

D I S A I M I S : 

OMAIN NTERCEPTS YMMETRY SYMPTOTES NTERVALS AX MIN NFLECTION KETCH D I S A I M I S Basic elements of Curve sketching (In A Nutshell )

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x y O (0, 1) Step Two x- intercepts: y = 0  x  .7 and  2.7 Range: x = ±   y =  .7 2.7 - .7 -2.7 CURVE SKETCHING ? + + + + + + + + + + + + + + -2 2 -3 Step One Domain: x : -  to +  0 y-intercept x = 0  y = 1 -  + 

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M C S C

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COUNCIL S O C I E T Y Mathematics CENTRE

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Thanks Bye