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LESSON PLAN 42

CARTESIAN PRODUCT OF SETS

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CARTESIAN PRODUCT

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PROPERTIES OF CARTESIAN PRODUCT

  • Two ordered pairs are equal iff the corresponding first elements are equal and the second elements are equal.
  • If any of the sets A or B is null set, then A x B = Ø.
  • If A and B are non empty sets and either A or B is infinite, so is A x B.
  • If n(A) = p and n(B) = q then n(AxB) = pq

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HOME ASSIGNMENT

  1. If (x+1, y-2) = (3,1) find x,y
  2. If P = {1,2} find P x P x P

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LESSON PLAN 43

RELATIONS

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RELATION

  • A relation R from a non empty set A to non empty set B is the subset of A x B where there is relation between the first and second element of ordered pair

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  • DOMAIN : The set off all first elements of the ordered pair is a relation R from a set A to set B is called Domain.
  • RANGE : The set of all second elements in a relation R from a to set A to a set B is called Range.
  • CODOMAIN : The set B is called the Codomain of R

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REPRESENTATION OF SETS

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NUMBER OF RELATIONS

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HOME ASSIGNMENT

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LESSON PLAN 44

FUNCTIONS

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FUNCTION

  • A relation f from a set A to a set B is called to be a function if every element of A has one image set A

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IMAGE AND PRE IMAGE

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REAL FUNCTIONS

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HOME ASSIGNMENTS

  1. Which are functions?
    1. R = {(2,1) , (3,1) , (4,2)}
    2. R = {(2,2) , (2,4) , (3,3) , (3,4)}

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LESSON PLAN 47

ALGEBRA OF REAL FUNCTIONS

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ADDITION OF REAL FUNCTIONS

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SUBTRACTION OF REAL FUNCTIONS

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MULTIPLICATION BY SCALAR

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PRODUCT OF REAL FUNCTIONS

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DIVISION OF REAL FUNCTIONS

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HOME ASSIGNMENT

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LESSON PLAN 48

TYPES OF RELATIONS

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EMPTY RELATION

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UNIVERSAL RELATION

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REFLEXIVE RELATION

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SYMMETRIC RELATION

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TRANSITIVE RELATION

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HOME ASSIGNMENT

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LESSON PLAN 49

EQUIVALENCE RELATION AND EQUIVALENCE CLASS

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EQUIVALENCE RELATION

A relation R on a set A is said to be equivalence relation if R is reflexive, symmetric and transitive.

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EQUIVALENCE CLASS

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HOME ASSIGNMENT

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LESSON PLAN 50

ONE-ONE, ONTO AND BIJECTIVE FUNCTIONS

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FUNCTION

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ONE-ONE (INJECTIVE) FUNCTION

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ONTO(SURJECTIVE) FUNCTION

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BIJECTIVE FUNCTION

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HOME ASSIGNMENT

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LESSON PLAN 51

CHARACTERISTIC DIFFERENCE BETWEEN A FINITE AND AN INFINITE SET.

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CHARACTERISATION OF FINITE SETS

Any one-one function on a finite set is onto and any onto function on a finite set is one one.

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HOME ASSIGNMENT

  1. Show that

a) Modulus function

b) Greatest integer function

Is neither one-one nor onto.

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LESSON PLAN 52

COMPOSITION OF FUNCTIONS

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COMPOSITION OF FUNCTIONS

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PRINCIPLES

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HOME ASSIGNMENT

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LESSON PLAN 53

  • DETERMINING ONE-ONE, ONTO AND BIJECTIVE FUNCTIONS USING THEIR GRAPHS.
  • RESULTS ON COMPOSITION OF FUNCTIONS

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PRINCIPLES

  • Any horizontal line drawn on the graph intersects the graph of a one-one function at most once.
  • Any horizontal line drawn on the graph intersects the graph of an onto function at least once.
  • Any horizontal line drawn on the graph intersects the graph of a bijective function exactly once.

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RESULTS ON FUNCTION COMPOSITION

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HOME ASSIGNMENT

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LESSON PLAN 54

INVERTIBLE FUNCTIONS

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INVERTIBLE FUNCTIONS

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INVERTIBILITY OF A FUNCTION

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HOME ASSIGNMENT

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LESSON PLAN 55

RESULTS ON INVERSE OF FUNCTIONS AND COMPOSITION OF FUNCTIONS

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ASSOCIATIVITY OF FUNCTION COMPOSITION

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INVERTIBILITY OF FUNCTION COMPOSITION

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HOME ASSIGNMENT

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LESSON PLAN 56

BINARY OPERATIONS

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FUNCTION

  • A relation f from a set A to a set B is said to be a function if every element of A has unique image in B.

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BINARY OPERATION

  • A binary operation * on a set ‘A’ is a function * : A x A🡪 A.

We denote *(a,b) by a*b.

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HOME ASSIGNMENT

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LESSON PLAN 57

PROPERTIES OF BINARY OPERATIONS

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BINARY OPERATION

  • A binary operation * on a set ‘A’ is a function * : A x A🡪 A.

We denote *(a,b) by a*b.

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COMMUTATIVITY OF BINARY OPERATION

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ASSOCIATIVITY OF BINARY OPERATION

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EXISTENCE OF IDENTITY

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EXISTENCE OF INVERSE

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LESSON PLAN 58

VOLUME OF PRISM

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PRISMS

  • Three dimensional objects whose surface is made up of two identical polygons are rectangles of same height, with the identical polygons on opposite sides is Prism.

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VOLUME OF PRISM

  • The volume of any prism is the product of base area and height.

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HOME ASSIGNMENT

  • The base of a prism is an equilateral triangle with perimeter 21cm and height 10cm. Find volume.

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LESSON PLAN 59

SURFACE AREA OF PRISM

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LATERAL SURFACE AREA OF PRISM

  • Lateral surface area = Perimeter of base * Height

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TOTAL SURFACE AREA OF PRISM

  • Total surface area = Lateral surface area + (2 * Area of base)

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LESSON PLAN 60

SURFACE AREA OF CYLINDER

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CURVED SURFACE AREA OF CYLINDER

  • Curved surface area = Perimeter of base * Height

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TOTAL SURFACE AREA OF CYLINDER

  • Total surface area = Curved surface area of cylinder + 2 * (Area of base)

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HOME ASSIGNMENT

  • Rahul’s house has a well of radius 1m and depth 10m. He wishes to plaster it. How much area should he plaster?

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