Sets

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Sets

  1. 1. Set Theory Sets, elements, kinds of numbers
  2. 2. Sets <ul><li>Set- A collection (usually of numbers) </li></ul><ul><ul><li>People tend to want to group things. A set is amthematical way to group things, usually numbers </li></ul></ul><ul><li>Elements / Members </li></ul><ul><ul><li>The things (numbers) that make up a set </li></ul></ul>
  3. 3. Sets <ul><li>3 ways to describe a set </li></ul><ul><ul><li>Word description </li></ul></ul><ul><ul><ul><li>All the odd numbers between 11 and 35 </li></ul></ul></ul><ul><ul><li>Listing Method </li></ul></ul><ul><ul><ul><li>{ 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33} </li></ul></ul></ul><ul><ul><ul><ul><li>Notice the { } symbols. The denote a SET </li></ul></ul></ul></ul><ul><ul><ul><ul><li>Commas are essential </li></ul></ul></ul></ul><ul><ul><li>Set-Builder Notation </li></ul></ul><ul><ul><ul><ul><li>{x|x is an odd number between 11 and 35 } </li></ul></ul></ul></ul><ul><ul><ul><ul><li>X such that x is an odd number… </li></ul></ul></ul></ul>
  4. 4. Sets <ul><li>Set-builder notation </li></ul><ul><ul><li>Uses a variable, usually x </li></ul></ul><ul><ul><ul><li>The variable before the line represents elements of a set, in general </li></ul></ul></ul><ul><ul><ul><li>After line- state criteria for the set </li></ul></ul></ul><ul><ul><li>Sets are given names for instance if we use E </li></ul></ul><ul><ul><ul><li>E= {1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 35, 27, 29, 31, 33, 35, 37, 39, 41} </li></ul></ul></ul>
  5. 5. Sets <ul><li>Set notation can be shortened by establishing a pattern, using ellipses </li></ul><ul><ul><li>E= {1, 3, 5, 7, ……….. 39, 41} </li></ul></ul><ul><li>A set with nothing (no elements) </li></ul><ul><ul><li>Empty set or null set </li></ul></ul><ul><ul><li>Notated by ø or {} </li></ul></ul><ul><ul><ul><li>Not {ø} this is a set with one element; ø </li></ul></ul></ul>
  6. 6. Elements of Sets <ul><li>q ϵ E </li></ul><ul><ul><li>q is an element of e </li></ul></ul><ul><li>Are the following statements true? </li></ul><ul><ul><li>3 ϵ {1, 2, 5, 9, 13} </li></ul></ul><ul><ul><li>0 ϵ {0, 1, 2, 3} </li></ul></ul><ul><ul><li>0.5 ϵ {0.2, 0.5, 0.8}   </li></ul></ul><ul><ul><ul><li>No </li></ul></ul></ul><ul><ul><ul><li>Yes </li></ul></ul></ul><ul><ul><ul><li>Yes </li></ul></ul></ul>
  7. 7. Elements of Sets <ul><li>The number of elements in a set is called the Cardinal Number or Cardinality of the set  </li></ul><ul><ul><li>n(A) </li></ul></ul><ul><ul><ul><li>Read as n of A </li></ul></ul></ul><ul><ul><li>Means number of elements in set A </li></ul></ul><ul><ul><li>Find n(D) </li></ul></ul><ul><ul><ul><li>D= {3, 7, 26, 678, 920} </li></ul></ul></ul><ul><ul><ul><li>N(D)= 5 </li></ul></ul></ul>
  8. 8. Elements of Sets <ul><li>If the same element is listed more than once in a set, you don’t count them more than once when finding the cardinality of a set </li></ul><ul><ul><li>Find n (R) </li></ul></ul><ul><ul><li>R= {0,1, 1, 2, 2, 3, 3} </li></ul></ul><ul><ul><li>N(R) = 4 </li></ul></ul>
  9. 9. Elements of Sets <ul><li>Sets can either be infinite or finite </li></ul><ul><ul><li>There can be an amount that could eventually be counted. These are Finite Sets </li></ul></ul><ul><ul><ul><li>We can find the Cardinal number for these sets </li></ul></ul></ul><ul><ul><li>Numbers go on forever without ending. Therefore, some sets can go on forever. These are Infinite Sets </li></ul></ul><ul><ul><ul><li>It is not possible to find a cardinal number for infinite sets </li></ul></ul></ul><ul><ul><ul><ul><li>Why can’t we find a cardinal number for infinite sets? </li></ul></ul></ul></ul>

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