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Set concepts

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Set theory in Algebra Mathematics

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Set concepts

  1. 1. Set Concepts
  2. 2. Introduction
  3. 3. <ul><li>Shaan Education society’s </li></ul><ul><li>Guardian college of education </li></ul><ul><li>Technology based lesson </li></ul>
  4. 4. <ul><li>NAME OF THE STUDENT: MALTI RAI </li></ul><ul><li>NAME OF GUIDE: MRS NILOFER MOMIN </li></ul><ul><li>NAME OF THE INCHARGE: MRS LEENA CHOUDHR Y </li></ul>
  5. 5. <ul><li>SUBJECT: Mathematics </li></ul><ul><li>UNIT : Set concepts </li></ul><ul><li>STANDARD: IX </li></ul>
  6. 6. INDEX <ul><li>Objectives </li></ul><ul><li>Definition of set </li></ul><ul><li>Properties of sets </li></ul><ul><li>Set theory </li></ul><ul><li>Venn Diagram </li></ul><ul><li>Set Representation </li></ul><ul><li>Types of Sets </li></ul><ul><li>Operation on Sets </li></ul>
  7. 7. <ul><li>Understanding set theory helps people to … </li></ul><ul><li>see things in terms of systems </li></ul><ul><li>organize things into groups </li></ul><ul><li>begin to understand logic </li></ul>Objectives
  8. 8. Definition of set <ul><li>A set is a well defined collection of objects. </li></ul><ul><li>Individual objects in set are called as elements of set. </li></ul><ul><li>e. g. 1. Collection of even numbers between 10 and 20. </li></ul><ul><li>2. Collection of flower or bouquet. </li></ul>
  9. 9. Properties of Sets <ul><li>1 Sets are denoted by capital letters. </li></ul><ul><li>Set notation : A ,B, C ,D </li></ul><ul><li>Elements of set are denoted by small letters. </li></ul><ul><li>Element notation : a,d,f,g, </li></ul><ul><li>For example SetA= {x,y,v,b,n,h,} </li></ul>
  10. 10. <ul><li>3 If x is element of A we can write as </li></ul><ul><li>x  A i.e x belongs to set A. </li></ul><ul><li>4. If x is not an element of A we can write as </li></ul><ul><li>x  A i.e x does not belong to A </li></ul><ul><li>e.g If Y is a set of days in a week then </li></ul><ul><li> Monday  A </li></ul><ul><li>and January  A </li></ul><ul><li>      </li></ul>
  11. 11. <ul><li>5 Each element is written once. </li></ul><ul><li>6 Set of Natural no. represented by- N , Whole no by- W , Integers by – I , Rational no by- Q , Real no by- R </li></ul><ul><li>7 Order of element is not important. </li></ul><ul><li>i.e set A can be written as </li></ul><ul><li>{ 1,2,3,4,5,} or as {5,2,3,4,1} </li></ul><ul><li>There is no difference between two. </li></ul>
  12. 12. Set Theory <ul><li>Georg cantor a German Mathematician born in Russia is creator of set theory </li></ul><ul><li>The concept of infinity was developed by cantor. </li></ul><ul><li>Proved real no. are more numerous than natural numbers. </li></ul><ul><li>Defined cardinal and ordinal no. </li></ul>Georg cantor
  13. 13. Venn Diagrams <ul><li>Born in 1834 in England. </li></ul><ul><li>Devised a simple diagramatic way to represent sets. </li></ul><ul><li>Here set are represented by closed figures such as : </li></ul>John Venn .a .i .g .y .2 .6 .8
  14. 14. Set Representation <ul><li>There are two main ways of representing sets. </li></ul><ul><li>Roaster method or Tabular method. </li></ul><ul><li>Set builder method or Rule method </li></ul>
  15. 15. Roster or Listing method <ul><li>All elements of the sets are listed,each element separated by comma(,) and enclosed within brackets </li></ul>
  16. 16. Roster or Listing method <ul><li>All elements of the sets are listed,each element separated by comma(,) and enclosed within brackets { } </li></ul><ul><li>e.g Set C= {1,6,8,4} </li></ul><ul><li>Set T ={Monday,Tuesdy,Wednesday,Thursday,Friday,Saturday} </li></ul><ul><li>Set k={a,e,i,o,u} </li></ul>
  17. 17. Rule method or set builder method <ul><li>All elements of set posses a common property </li></ul><ul><li>e.g. set of natural numbers is represented by </li></ul><ul><li>K= {x|x is a natural no} </li></ul><ul><li>Here | stands for ‘such that’ </li></ul><ul><li>‘ :’ can be used in place of ‘|’ </li></ul><ul><li>e.g. Set T={y|y is a season of the year} </li></ul><ul><li>Set H={x|x is blood type} </li></ul>
  18. 18. Cardianility of set <ul><li>Number of element in a set is called as cardianility of set. </li></ul><ul><li>No of elements in set n (A) </li></ul><ul><li>e.g Set A= {he,she, it,the, you} </li></ul><ul><li>Here no. of elements are n |A|=5 </li></ul><ul><li>Singleton set containing only one elements e.g Set A={3} </li></ul>
  19. 19. Types of set <ul><li>Empty set </li></ul><ul><li>Finite set </li></ul><ul><li>Infinite set </li></ul><ul><li>Equal set </li></ul><ul><li>Equivalent set </li></ul><ul><li>Subset Universal set </li></ul>
  20. 20. Equal sets <ul><li>Two sets k and R are called equal if they have equal numbers and of similar types of elements. </li></ul><ul><li>For e.g. If k={1,3,4,5,6} </li></ul><ul><li>R={1,3,4,5,6} then both Set k and R are equal. </li></ul><ul><li>We can write as Set K=Set R </li></ul>
  21. 21. Empty sets <ul><li>A set which does not contain any elements is called as Empty set or Null or Void set. Denoted by  or { } </li></ul><ul><li>e.g. Set A= {set of months containing </li></ul><ul><li>32 days} </li></ul><ul><li>Here n (A)= 0; hence A is an empty set. </li></ul><ul><li>e.g. set H={no of cars with three wheels} </li></ul><ul><li>Here n (H)= 0; hence it is an empty set. </li></ul>
  22. 22. Finite set <ul><li>Set which contains definite no of element. </li></ul><ul><li>e.g. Set A= {  ,  ,  ,  } </li></ul><ul><li>Counting of elements is fixed. </li></ul><ul><li>Set B = { x|x is no of pages in a particular book} </li></ul><ul><li>Set T ={ y|y is no of seats in a bus} </li></ul>
  23. 23. Infinite set <ul><li>A set which contains indefinite numbers of elements. </li></ul><ul><li>Set A= { x|x is a of whole numbers} </li></ul><ul><li>Set B = {y|y is point on a line} </li></ul>
  24. 24. Subset <ul><li>Sets which are the part of another set are called subsets of the original set. For example, if A={3,5,6,8} and B ={1,4,9} then B is a subset of A it is represented as B  A </li></ul><ul><li>Every set is subset of itself i.e A  A </li></ul><ul><li>Empty set is a subset of every set. i.e  A </li></ul>.3 .5 .6. .8 .1 .4 .9 A B
  25. 25. Universal set <ul><li>The universal set is the set of all elements pertinent to a given discussion It is designated by the symbol U </li></ul><ul><li>e.g. Set T ={The deck of ordinary playing cards}. Here each card is an element of universal set. </li></ul><ul><li>Set A= {All the face cards} </li></ul><ul><li>Set B= {numbered cards} </li></ul><ul><li>Set C= {Poker hands} each of these sets are Subset of universal set T </li></ul>
  26. 26. Operation on Sets <ul><li>Intersection of sets </li></ul><ul><li>Union of sets </li></ul><ul><li>Difference of two sets </li></ul><ul><li>Complement of a set </li></ul>
  27. 27. Intersection of sets <ul><li>Let A and B be two sets. Then the set of all common elements of A and B is called the Intersection of A and B and is denoted by A ∩B </li></ul><ul><li>Let A={1,2,3,7,11,13}} </li></ul><ul><ul><li> B={1,7,13,4,10,17}} </li></ul></ul><ul><li>Then a set C= {1,7,13}} contains the elements common to both A and B </li></ul><ul><li>Hence A∩B is represented by shaded part in venn diagram . </li></ul><ul><li>Thus A∩B={x|x  A and x  B} </li></ul>
  28. 28. Union of sets <ul><li>Let A and B be two given sets then the set of all elements which are in the set A or in the set B is called the union of two sets and is denoted by AUB and is read as ‘A union B’ </li></ul><ul><li>Union of Set A= {1, 2, 3, 4, = {0, 2, 4, 6 } </li></ul><ul><li>5, 6} and Set B </li></ul>
  29. 29. Difference of two sets <ul><ul><li>The difference of set A- B is set of all elements of “A” which does not belong to “B”. </li></ul></ul><ul><ul><li>In set builder form difference of set is:- </li></ul></ul><ul><ul><li>A-B= {x: x  A x  B} </li></ul></ul><ul><ul><li>B-A={x: x  B x  A} </li></ul></ul><ul><ul><li>e.g SetA ={ 1,4,7,8,9} </li></ul></ul><ul><ul><li>Set B= {3,2,1,7,5} </li></ul></ul><ul><ul><li>Then A-B = { 4,8,9} </li></ul></ul>
  30. 30. Disjoint sets <ul><li>Sets that have no common members are called disjoint sets. </li></ul><ul><li>Example: Given that </li></ul><ul><li>U= {1,2,3,4,5,6,7,8,9,10} </li></ul><ul><li>setA={ 1,2,3,4,5} </li></ul><ul><li>setC={ 8,10} </li></ul><ul><li>No common elements hence set A and are disjoint set. </li></ul>
  31. 31. Summarisation <ul><li>Definition of set andProperties of sets </li></ul><ul><li>Set theory </li></ul><ul><li>Venn Diagram </li></ul><ul><li>Set Representation </li></ul><ul><li>Types of Sets </li></ul>
  32. 32. Home work <ul><li>1 Write definition of set concepts. </li></ul><ul><li>2 What is intersection and union of sets. </li></ul><ul><li>3 Explain properties of sets with examples. </li></ul>
  33. 33. Applications <ul><li>A set having no element is empty set. </li></ul><ul><li>( yes / no ) </li></ul><ul><li>2.A set having only one element is singleton set. ( yes / no ) </li></ul><ul><li>3.A set containing fixed no of elements.{ finite / infinite set ) </li></ul><ul><li>4.Two set having no common element. ( disjoint set / complement set ) </li></ul>
  34. 34. <ul><li>Yes your answer is right </li></ul>
  35. 35. <ul><li>No is wrong your answer </li></ul>
  36. 36. <ul><li>Yes your answer is right </li></ul>
  37. 37. <ul><li>No your answer is wrong </li></ul>
  38. 38. <ul><li>Yes your answer is right </li></ul>
  39. 39. <ul><li>No your answer is wrong </li></ul>
  40. 40. <ul><li>Yes your answer is right </li></ul>
  41. 41. <ul><li>No your answer is wrong </li></ul>

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