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### 3 6 introduction to sets-optional

1. 1. Introduction to Sets<br /> (Optional) <br />Frank Ma © 2011<br />
2. 2.
3. 3. Introduction to Sets<br />A setis a backpack which may or may not contains any items. <br />
4. 4. Introduction to Sets<br />A setis a backpack which may or may not contains any items. <br />We use capital letters as names of sets. <br />
5. 5. Introduction to Sets<br />A setis a backpack which may or may not contains any items. <br />We use capital letters as names of sets. For example, <br />A = { } ≡ Ф (phi) – which is the empty set<br />B = {wallet, my car–key, math–book} <br />C = {my car–key}<br />D = {1, 2, 3, ….} = {all positive integer}<br />
6. 6. Introduction to Sets<br />A setis a backpack which may or may not contains any items. <br />We use capital letters as names of sets. For example, <br />A = { } ≡ Ф (phi) – which is the empty set<br />B = {wallet, my car–key, math–book} <br />C = {my car–key}<br />D = {1, 2, 3, ….} = {all positive integer}<br />Note that we use “{ }” to symbolize the backpack and also that a set may contain infinitely many items. <br />
7. 7. Introduction to Sets<br />A setis a backpack which may or may not contains any items. <br />We use capital letters as names of sets. For example, <br />A = { } ≡ Ф (phi) – which is the empty set<br />B = {wallet, my car–key, math–book} <br />C = {my car–key}<br />D = {1, 2, 3, ….} = {all positive integer}<br />Note that we use “{ }” to symbolize the backpack and also that a set may contain infinitely many items. <br />Each item of a set is called an element of that set. <br />
8. 8. Introduction to Sets<br />A setis a backpack which may or may not contains any items. <br />We use capital letters as names of sets. For example, <br />A = { } ≡ Ф (phi) – which is the empty set<br />B = {wallet, my car–key, math–book} <br />C = {my car–key}<br />D = {1, 2, 3, ….} = {all positive integer}<br />Note that we use “{ }” to symbolize the backpack and also that a set may contain infinitely many items. <br />Each item of a set is called an element of that set. <br />We write x ϵ S if x is an element the set S. <br />
9. 9. Introduction to Sets<br />A setis a backpack which may or may not contains any items. <br />We use capital letters as names of sets. For example, <br />A = { } ≡ Ф (phi) – which is the empty set<br />B = {wallet, my car–key, math–book} <br />C = {my car–key}<br />D = {1, 2, 3, ….} = {all positive integer}<br />Note that we use “{ }” to symbolize the backpack and also that a set may contain infinitely many items. <br />Each item of a set is called an element of that set. <br />We write x ϵ S if x is an element the set S. <br />Hence from the above sets we’ve that<br />my car–key<br />ϵ B<br />
10. 10. Introduction to Sets<br />A setis a backpack which may or may not contains any items. <br />We use capital letters as names of sets. For example, <br />A = { } ≡ Ф (phi) – which is the empty set<br />B = {wallet, my car–key, math–book} <br />C = {my car–key}<br />D = {1, 2, 3, ….} = {all positive integer}<br />Note that we use “{ }” to symbolize the backpack and also that a set may contain infinitely many items. <br />Each item of a set is called an element of that set. <br />We write x ϵ S if x is an element the set S. <br />Hence from the above sets we’ve that<br />my car–key<br />ϵ B<br />my car–key<br />ϵ C<br />
11. 11. Introduction to Sets<br />A setis a backpack which may or may not contains any items. <br />We use capital letters as names of sets. For example, <br />A = { } ≡ Ф (phi) – which is the empty set<br />B = {wallet, my car–key, math–book} <br />C = {my car–key}<br />D = {1, 2, 3, ….} = {all positive integer}<br />Note that we use “{ }” to symbolize the backpack and also that a set may contain infinitely many items. <br />Each item of a set is called an element of that set. <br />We write x ϵ S if x is an element the set S. <br />Hence from the above sets we’ve that<br />my car–key<br />ϵ A<br />my car–key<br />ϵ B<br />my car–key<br />ϵ C<br />
12. 12. Introduction to Sets<br />A setis a backpack which may or may not contains any items. <br />We use capital letters as names of sets. For example, <br />A = { } ≡ Ф (phi) – which is the empty set<br />B = {wallet, my car–key, math–book} <br />C = {my car–key}<br />D = {1, 2, 3, ….} = {all positive integer}<br />Note that we use “{ }” to symbolize the backpack and also that a set may contain infinitely many items. <br />Each item of a set is called an element of that set. <br />We write x ϵ S if x is an element the set S. <br />Hence from the above sets we’ve that<br />my car–key<br />ϵ A<br />my car–key<br />ϵ B<br />my car–key<br />ϵ C<br />Note that C is not an element of B because C is just another<br />backpack whose content part (or all) of B’s content. <br />
13. 13. Introduction to Sets<br />A setis a backpack which may or may not contains any items. <br />We use capital letters as names of sets. For example, <br />A = { } ≡ Ф (phi) – which is the empty set<br />B = {wallet, my car–key, math–book} <br />C = {my car–key}<br />D = {1, 2, 3, ….} = {all positive integer}<br />Note that we use “{ }” to symbolize the backpack and also that a set may contain infinitely many items. <br />Each item of a set is called an element of that set. <br />We write x ϵ S if x is an element the set S. <br />Hence from the above sets we’ve that<br />my car–key<br />ϵ A<br />my car–key<br />ϵ B<br />my car–key<br />ϵ C<br />Note that C is not an element of B because C is just another<br />backpack whose content part (or all) of B’s content. In general<br />T is a subset of S if every element of T is also an element of S and we write this asT S. <br />
14. 14. Introduction to Sets<br />A setis a backpack which may or may not contains any items. <br />We use capital letters as names of sets. For example, <br />A = { } ≡ Ф (phi) – which is the empty set<br />B = {wallet, my car–key, math–book} <br />C = {my car–key}<br />D = {1, 2, 3, ….} = {all positive integer}<br />Note that we use “{ }” to symbolize the backpack and also that a set may contain infinitely many items. <br />Each item of a set is called an element of that set. <br />We write x ϵ S if x is an element the set S. <br />Hence from the above sets we’ve that<br />my car–key<br />ϵ A<br />my car–key<br />ϵ B<br />my car–key<br />ϵ C<br />Note that C is not an element of B because C is just another<br />backpack whose content part (or all) of B’s content. In general<br />T is a subset of S if every element of T is also an element of S and we write this asT S. So we’ve C but C B.<br />ϵ B<br />
15. 15. Introduction to Sets<br />Two sets S and T are the same if they have exactly the same content, that is, every element in S is also an element of T and vice versa. <br />
16. 16. Introduction to Sets<br />Two sets S and T are the same if they have exactly the same content, that is, every element in S is also an element of T and vice versa. Hence if S = {a, b} and that T = {b ,a} then S = T.<br />
17. 17. Introduction to Sets<br />Two sets S and T are the same if they have exactly the same content, that is, every element in S is also an element of T and vice versa. Hence if S = {a, b} and that T = {b ,a} then S = T.<br />Note that the order of the elements is not important. <br />
18. 18. Introduction to Sets<br />Two sets S and T are the same if they have exactly the same content, that is, every element in S is also an element of T and vice versa. Hence if S = {a, b} and that T = {b ,a} then S = T.<br />Note that the order of the elements is not important. <br />Some sets of numbers may be represented graphically. <br />
19. 19. Introduction to Sets<br />Two sets S and T are the same if they have exactly the same content, that is, every element in S is also an element of T and vice versa. Hence if S = {a, b} and that T = {b ,a} then S = T.<br />Note that the order of the elements is not important. <br />Some sets of numbers may be represented graphically. <br />Example A. Graph the following sets.<br />a. A = {all the x where –2 < x ≤ 6}<br />
20. 20. Introduction to Sets<br />Two sets S and T are the same if they have exactly the same content, that is, every element in S is also an element of T and vice versa. Hence if S = {a, b} and that T = {b ,a} then S = T.<br />Note that the order of the elements is not important. <br />Some sets of numbers may be represented graphically. <br />Example A. Graph the following sets.<br />a. A = {all the x where –2 < x ≤ 6}<br />x<br />–2<br />6<br />
21. 21. Introduction to Sets<br />Two sets S and T are the same if they have exactly the same content, that is, every element in S is also an element of T and vice versa. Hence if S = {a, b} and that T = {b ,a} then S = T.<br />Note that the order of the elements is not important. <br />Some sets of numbers may be represented graphically. <br />Example A. Graph the following sets.<br />a. A = {all the x where –2 < x ≤ 6}<br />x<br />–2<br />6<br />b. B = {all the x where | x | ≤ 4}<br />
22. 22. Introduction to Sets<br />Two sets S and T are the same if they have exactly the same content, that is, every element in S is also an element of T and vice versa. Hence if S = {a, b} and that T = {b ,a} then S = T.<br />Note that the order of the elements is not important. <br />Some sets of numbers may be represented graphically. <br />Example A. Graph the following sets.<br />a. A = {all the x where –2 < x ≤ 6}<br />x<br />–2<br />6<br />b. B = {all the x where | x | ≤ 4}<br />This is the same as { x where –4 ≤ x ≤ 4}<br />
23. 23. Introduction to Sets<br />Two sets S and T are the same if they have exactly the same content, that is, every element in S is also an element of T and vice versa. Hence if S = {a, b} and that T = {b ,a} then S = T.<br />Note that the order of the elements is not important. <br />Some sets of numbers may be represented graphically. <br />Example A. Graph the following sets.<br />a. A = {all the x where –2 < x ≤ 6}<br />x<br />–2<br />6<br />b. B = {all the x where | x | ≤ 4}<br />This is the same as { x where –4 ≤ x ≤ 4}<br />x<br />–4<br />4<br />
24. 24. Introduction to Sets<br />Two sets S and T are the same if they have exactly the same content, that is, every element in S is also an element of T and vice versa. Hence if S = {a, b} and that T = {b ,a} then S = T.<br />Note that the order of the elements is not important. <br />Some sets of numbers may be represented graphically. <br />Example A. Graph the following sets.<br />a. A = {all the x where –2 < x ≤ 6}<br />x<br />–2<br />6<br />b. B = {all the x where | x | ≤ 4}<br />This is the same as { x where –4 ≤ x ≤ 4}<br />x<br />–4<br />4<br />Intersection and Union<br />
25. 25. Introduction to Sets<br />Two sets S and T are the same if they have exactly the same content, that is, every element in S is also an element of T and vice versa. Hence if S = {a, b} and that T = {b ,a} then S = T.<br />Note that the order of the elements is not important. <br />Some sets of numbers may be represented graphically. <br />Example A. Graph the following sets.<br />a. A = {all the x where –2 < x ≤ 6}<br />x<br />–2<br />6<br />b. B = {all the x where | x | ≤ 4}<br />This is the same as { x where –4 ≤ x ≤ 4}<br />x<br />–4<br />4<br />Intersection and Union<br />Let S and T be two sets, S T, read as S intersects T, is the set of common elements of S and T. <br />
26. 26. Introduction to Sets<br />Two sets S and T are the same if they have exactly the same content, that is, every element in S is also an element of T and vice versa. Hence if S = {a, b} and that T = {b ,a} then S = T.<br />Note that the order of the elements is not important. <br />Some sets of numbers may be represented graphically. <br />Example A. Graph the following sets.<br />a. A = {all the x where –2 < x ≤ 6}<br />x<br />–2<br />6<br />b. B = {all the x where | x | ≤ 4}<br />This is the same as { x where –4 ≤ x ≤ 4}<br />x<br />–4<br />4<br />Intersection and Union<br />Let S and T be two sets, S T, read as S intersects T, is the set of common elements of S and T. Hence {a, b} {b, c} = {b}.<br />
27. 27. Introduction to Sets<br />Two sets S and T are the same if they have exactly the same content, that is, every element in S is also an element of T and vice versa. Hence if S = {a, b} and that T = {b ,a} then S = T.<br />Note that the order of the elements is not important. <br />Some sets of numbers may be represented graphically. <br />Example A. Graph the following sets.<br />a. A = {all the x where –2 < x ≤ 6}<br />x<br />–2<br />6<br />b. B = {all the x where | x | ≤ 4}<br />This is the same as { x where –4 ≤ x ≤ 4}<br />x<br />–4<br />4<br />Intersection and Union<br />Let S and T be two sets, S T, read as S intersects T, is the set of common elements of S and T. Hence {a, b} {b, c} = {b}.<br />Note that the intersection is a set, not just the element “b”.<br />
28. 28. Introduction to Sets<br />Let S and T be two sets, S U T , read as S unions T, is the merge–set of the two sets. <br />
29. 29. Introduction to Sets<br />Let S and T be two sets, S U T , read as S unions T, is the merge–set of the two sets. For the union operation, duplicated items is only counted once.<br />
30. 30. Introduction to Sets<br />Let S and T be two sets, S U T , read as S unions T, is the merge–set of the two sets. For the union operation, duplicated items is only counted once. Hence {a, b} U {b, c} = {a, b, c}.<br />
31. 31. Introduction to Sets<br />Let S and T be two sets, S U T , read as S unions T, is the merge–set of the two sets. For the union operation, duplicated items is only counted once. Hence {a, b} U {b, c} = {a, b, c}.<br />Example B. Find and draw A B and A U B given that<br />A = {all the x where –2 < x ≤ 6}<br />B = {all the x where | x | ≤ 4}<br />
32. 32. Introduction to Sets<br />Let S and T be two sets, S U T , read as S unions T, is the merge–set of the two sets. For the union operation, duplicated items is only counted once. Hence {a, b} U {b, c} = {a, b, c}.<br />Example B. Find and draw A B and A U B given that<br />A = {all the x where –2 < x ≤ 6}<br />B = {all the x where | x | ≤ 4}<br />We have graphically<br />A<br />x<br />–4<br />4<br />B<br />x<br />–2<br />6<br />
33. 33. Introduction to Sets<br />Let S and T be two sets, S U T , read as S unions T, is the merge–set of the two sets. For the union operation, duplicated items is only counted once. Hence {a, b} U {b, c} = {a, b, c}.<br />Example B. Find and draw A B and A U B given that<br />A = {all the x where –2 < x ≤ 6}<br />B = {all the x where | x | ≤ 4}<br />We have graphically<br />A<br />x<br />–4<br />4<br />B<br />x<br />–2<br />6<br />A B <br />4<br />–2<br />A B is the common or overlapped segment of A with B.<br />
34. 34. Introduction to Sets<br />Let S and T be two sets, S U T , read as S unions T, is the merge–set of the two sets. For the union operation, duplicated items is only counted once. Hence {a, b} U {b, c} = {a, b, c}.<br />Example B. Find and draw A B and A U B given that<br />A = {all the x where –2 < x ≤ 6}<br />B = {all the x where | x | ≤ 4}<br />We have graphically<br />A<br />x<br />–4<br />4<br />B<br />x<br />–2<br />6<br />A B <br />4<br />–2<br />A B is the common or overlapped segment of A with B. <br />A = { B = {–2 < x ≤ 4}<br />
35. 35. Introduction to Sets<br />Given that<br />A<br />x<br />–4<br />4<br />B<br />x<br />–2<br />6<br />
36. 36. Introduction to Sets<br />Given that<br />A<br />x<br />–4<br />4<br />B<br />x<br />–2<br />6<br />A U B consists both portions of either color<br />
37. 37. Introduction to Sets<br />Given that<br />A<br />x<br />–4<br />4<br />B<br />x<br />–2<br />6<br />A U B <br />–4<br />6<br />A U B consists both portions of either color = {–4 < x ≤ 6}.<br />
38. 38. Introduction to Sets<br />Given that<br />A<br />x<br />–4<br />4<br />B<br />x<br />–2<br />6<br />A U B <br />–4<br />6<br />A U B consists both portions of either color = {–4 < x ≤ 6}.<br />Let’s extend this to the x&y coordinate system. <br />
39. 39. Introduction to Sets<br />Given that<br />A<br />x<br />–4<br />4<br />B<br />x<br />–2<br />6<br />A U B <br />–4<br />6<br />A U B consists both portions of either color = {–4 < x ≤ 6}.<br />Let’s extend this to the x&y coordinate system. The basic 2D regions are half–planes and strips. <br />
40. 40. Introduction to Sets<br />Given that<br />A<br />x<br />–4<br />4<br />B<br />x<br />–2<br />6<br />A U B <br />–4<br />6<br />A U B consists both portions of either color = {–4 < x ≤ 6}.<br />Let’s extend this to the x&y coordinate system. The basic 2D regions are half–planes and strips. <br />Example. C. Shade the following sets in the rectangular system. <br />a. A = {(x, y) where x > 0}<br />
41. 41. Introduction to Sets<br />Given that<br />A<br />x<br />–4<br />4<br />B<br />x<br />–2<br />6<br />A U B <br />–4<br />6<br />A U B consists both portions of either color = {–4 < x ≤ 6}.<br />Let’s extend this to the x&y coordinate system. The basic 2D regions are half–planes and strips. <br />Example. C. Shade the following sets in the rectangular system. <br />a. A = {(x, y) where x > 0}<br />Note that there is no mention of y means that y may take on any value.<br />
42. 42. Introduction to Sets<br />Given that<br />A<br />x<br />–4<br />4<br />B<br />x<br />–2<br />6<br />A U B <br />–4<br />6<br />A U B consists both portions of either color = {–4 < x ≤ 6}.<br />Let’s extend this to the x&y coordinate system. The basic 2D regions are half–planes and strips. <br />Example. C. Shade the following sets in the rectangular system. <br />a. A = {(x, y) where x > 0}<br />{(x, y) where x > 0}<br />Note that there is no mention of y means that y may take on any value. <br />
43. 43. Introduction to Sets<br />Given that<br />A<br />x<br />–4<br />4<br />B<br />x<br />–2<br />6<br />A U B <br />–4<br />6<br />A U B consists both portions of either color = {–4 < x ≤ 6}.<br />Let’s extend this to the x&y coordinate system. The basic 2D regions are half–planes and strips. <br />Example. C. Shade the following sets in the rectangular system. <br />a. A = {(x, y) where x > 0}<br />{(x, y) where x > 0}<br />Note that there is no mention of y means that y may take on any value. The open region is called ahalf–plane and the dash–line means exclusion. <br />
44. 44. Introduction to Sets<br />b. B = {(x, y) where 0 < y ≤ 4}<br />
45. 45. Introduction to Sets<br />b. B = {(x, y) where 0 < y ≤ 4}<br />There is no restriction on x <br />so x may take on any value.<br />
46. 46. Introduction to Sets<br />b. B = {(x, y) where 0 < y ≤ 4}<br />There is no restriction on x <br />so x may take on any value.<br />{(x, y) where 0 < y ≤ 4}<br />
47. 47. Introduction to Sets<br />b. B = {(x, y) where 0 < y ≤ 4}<br />There is no restriction on x <br />so x may take on any value.<br />The open region we obtained is called a strip.<br />{(x, y) where 0 < y ≤ 4}<br />
48. 48. Introduction to Sets<br />b. B = {(x, y) where 0 < y ≤ 4}<br />There is no restriction on x <br />so x may take on any value.<br />The open region we obtained is called a strip. Note that the solid represent inclusion.<br />{(x, y) where 0 < y ≤ 4}<br />
49. 49. Introduction to Sets<br />b. B = {(x, y) where 0 < y ≤ 4}<br />There is no restriction on x <br />so x may take on any value.<br />The open region we obtained is called a strip. Note that the solid represent inclusion.<br />{(x, y) where 0 < y ≤ 4}<br />c. Let C = {(x, y) where –3 < x < 3 and 0 < y ≤ 4} <br /> draw B C. <br />
50. 50. Introduction to Sets<br />b. B = {(x, y) where 0 < y ≤ 4}<br />There is no restriction on x <br />so x may take on any value.<br />The open region we obtained is called a strip. Note that the solid represent inclusion.<br />{(x, y) where 0 < y ≤ 4}<br />c. Let C = {(x, y) where –3 < x < 3 and 0 < y ≤ 4} <br /> draw B C. <br />This is the overlap of two strips.<br />
51. 51. Introduction to Sets<br />b. B = {(x, y) where 0 < y ≤ 4}<br />There is no restriction on x <br />so x may take on any value.<br />The open region we obtained is called a strip. Note that the solid represent inclusion.<br />{(x, y) where 0 < y ≤ 4}<br />c. Let C = {(x, y) where –3 < x < 3 and 0 < y ≤ 4} <br /> draw B C. <br />This is the overlap of two strips.<br />B is the same as the above.<br />B<br />
52. 52. Introduction to Sets<br />b. B = {(x, y) where 0 < y ≤ 4}<br />There is no restriction on x <br />so x may take on any value.<br />The open region we obtained is called a strip. Note that the solid represent inclusion.<br />{(x, y) where 0 < y ≤ 4}<br />c. Let C = {(x, y) where –3 < x < 3 and 0 < y ≤ 4} <br /> draw B C. <br />C<br />This is the overlap of two strips.<br />B is the same as the above.<br />C is the vertical strip where<br />–3 < x < 3. <br />B<br />
53. 53. Introduction to Sets<br />b. B = {(x, y) where 0 < y ≤ 4}<br />There is no restriction on x <br />so x may take on any value.<br />The open region we obtained is called a strip. Note that the solid represent inclusion.<br />{(x, y) where 0 < y ≤ 4}<br />c. Let C = {(x, y) where –3 < x < 3 and 0 < y ≤ 4} <br /> draw B C. <br />C<br />This is the overlap of two strips.<br />B is the same as the above.<br />C is the vertical strip where<br />–3 < x < 3. <br />The overlap is the rectangular region. <br />B<br />B C<br />
54. 54. Introduction to Sets<br />b. B = {(x, y) where 0 < y ≤ 4}<br />There is no restriction on x <br />so x may take on any value.<br />The open region we obtained is called a strip. Note that the solid represent inclusion.<br />{(x, y) where 0 < y ≤ 4}<br />c. Let C = {(x, y) where –3 < x < 3 and 0 < y ≤ 4} <br /> draw B C. <br />C<br />This is the overlap of two strips.<br />B is the same as the above.<br />C is the vertical strip where<br />–3 < x < 3. <br />The overlap is the rectangular region. Note that all the corners<br />are excluded. <br />B<br />B C<br />
55. 55. Introduction to Sets<br />b. B = {(x, y) where 0 < y ≤ 4}<br />There is no restriction on x <br />so x may take on any value.<br />The open region we obtained is called a strip. Note that the solid represent inclusion.<br />{(x, y) where 0 < y ≤ 4}<br />c. Let C = {(x, y) where –3 < x < 3 and 0 < y ≤ 4} <br /> draw B C. <br />C<br />This is the overlap of two strips.<br />B is the same as the above.<br />C is the vertical strip where<br />–3 < x < 3. <br />The overlap is the rectangular region. Note that all the corners<br />are excluded. <br />You Do: Label the corners.<br />B<br />B C<br />