Completeness Axiom                      Mathematics 4                       June 8, 2012Mathematics 4 ()       Completenes...
DefinitionsUpper Bound and Lower Bound of a SetUpper Bound of a SetAn upper bound (UB) of a set is a number greater than or...
DefinitionsUpper Bound and Lower Bound of a SetUpper Bound of a SetAn upper bound (UB) of a set is a number greater than or...
DefinitionsUpper Bound and Lower Bound of a SetUpper and Lower Bounds of a SetDetermine three upper bounds and three lower ...
DefinitionsUpper Bound and Lower Bound of a SetUpper and Lower Bounds of a SetDetermine three upper bounds and three lower ...
DefinitionsUpper Bound and Lower Bound of a SetUpper and Lower Bounds of a SetDetermine three upper bounds and three lower ...
DefinitionsLeast Upper Bound and Greatest Lower Bound of a SetLeast Upper Bound of a SetThe Least Upper Bound (LUB) of a se...
DefinitionsLeast Upper Bound and Greatest Lower Bound of a SetLeast Upper Bound of a SetThe Least Upper Bound (LUB) of a se...
ExamplesGLB and LUB                                                 GLB   LUB                 {1, 2, 3, 4}     Mathematics...
ExamplesGLB and LUB                                                 GLB   LUB                 {1, 2, 3, 4}                ...
ExamplesGLB and LUB                                         GLB   LUB           {1, 2, 3, 4}               Z+      {x | x ...
ExamplesGLB and LUB                                         GLB   LUB           {1, 2, 3, 4}               Z+      {x | x ...
ExamplesGLB and LUB                                             GLB   LUB           {x | 0 < x ≤ 3}     Mathematics 4 ()  ...
ExamplesGLB and LUB                                             GLB   LUB          {x | 0 < x ≤ 3}          {x | x2 − 3 < ...
ExamplesGLB and LUB                                             GLB   LUB         {x | 0 < x ≤ 3}        {x | x2 − 3 < 0} ...
ExamplesGLB and LUB                                             GLB   LUB         {x | 0 < x ≤ 3}        {x | x2 − 3 < 0} ...
ExamplesGLB and LUB                                             GLB   LUB        {x | 0 < x ≤ 3}       {x | x2 − 3 < 0}   ...
Homework 4Find the GLB and LUB of the following sets                                                        3x  1      {x2...
Upcoming SlideShare
Loading in …5
×

Completeness axiom

1,347 views

Published on

Published in: Technology, Education
0 Comments
0 Likes
Statistics
Notes
  • Be the first to comment

  • Be the first to like this

No Downloads
Views
Total views
1,347
On SlideShare
0
From Embeds
0
Number of Embeds
1
Actions
Shares
0
Downloads
47
Comments
0
Likes
0
Embeds 0
No embeds

No notes for slide

Completeness axiom

  1. 1. Completeness Axiom Mathematics 4 June 8, 2012Mathematics 4 () Completeness Axiom June 8, 2012 1/7
  2. 2. DefinitionsUpper Bound and Lower Bound of a SetUpper Bound of a SetAn upper bound (UB) of a set is a number greater than or equal to anyelement of the set S. Mathematics 4 () Completeness Axiom June 8, 2012 2/7
  3. 3. DefinitionsUpper Bound and Lower Bound of a SetUpper Bound of a SetAn upper bound (UB) of a set is a number greater than or equal to anyelement of the set S.Lower Bound of a SetAn lower bound (LB) of a set is a number less than or equal to anyelement of the set S. Mathematics 4 () Completeness Axiom June 8, 2012 2/7
  4. 4. DefinitionsUpper Bound and Lower Bound of a SetUpper and Lower Bounds of a SetDetermine three upper bounds and three lower bounds for the followingsets, if these exists: 1 {1, 2, 3, 4} Mathematics 4 () Completeness Axiom June 8, 2012 3/7
  5. 5. DefinitionsUpper Bound and Lower Bound of a SetUpper and Lower Bounds of a SetDetermine three upper bounds and three lower bounds for the followingsets, if these exists: 1 {1, 2, 3, 4} 2 Z+ Mathematics 4 () Completeness Axiom June 8, 2012 3/7
  6. 6. DefinitionsUpper Bound and Lower Bound of a SetUpper and Lower Bounds of a SetDetermine three upper bounds and three lower bounds for the followingsets, if these exists: 1 {1, 2, 3, 4} 2 Z+ 3 N − {1, 2, 3} Mathematics 4 () Completeness Axiom June 8, 2012 3/7
  7. 7. DefinitionsLeast Upper Bound and Greatest Lower Bound of a SetLeast Upper Bound of a SetThe Least Upper Bound (LUB) of a set is the smallest possible upperbound. Mathematics 4 () Completeness Axiom June 8, 2012 4/7
  8. 8. DefinitionsLeast Upper Bound and Greatest Lower Bound of a SetLeast Upper Bound of a SetThe Least Upper Bound (LUB) of a set is the smallest possible upperbound.Greatest Lower Bound of a SetThe Greatest Lower Bound (GLB) of a set is the largest possible lowerbound. Mathematics 4 () Completeness Axiom June 8, 2012 4/7
  9. 9. ExamplesGLB and LUB GLB LUB {1, 2, 3, 4} Mathematics 4 () Completeness Axiom June 8, 2012 5/7
  10. 10. ExamplesGLB and LUB GLB LUB {1, 2, 3, 4} Z+ Mathematics 4 () Completeness Axiom June 8, 2012 5/7
  11. 11. ExamplesGLB and LUB GLB LUB {1, 2, 3, 4} Z+ {x | x ∈ R, |x| < 4} Mathematics 4 () Completeness Axiom June 8, 2012 5/7
  12. 12. ExamplesGLB and LUB GLB LUB {1, 2, 3, 4} Z+ {x | x ∈ R, |x| < 4} N − {1, 2, 3} Mathematics 4 () Completeness Axiom June 8, 2012 5/7
  13. 13. ExamplesGLB and LUB GLB LUB {x | 0 < x ≤ 3} Mathematics 4 () Completeness Axiom June 8, 2012 6/7
  14. 14. ExamplesGLB and LUB GLB LUB {x | 0 < x ≤ 3} {x | x2 − 3 < 0} Mathematics 4 () Completeness Axiom June 8, 2012 6/7
  15. 15. ExamplesGLB and LUB GLB LUB {x | 0 < x ≤ 3} {x | x2 − 3 < 0} x {y | y = x+1 , x ≥ 0} Mathematics 4 () Completeness Axiom June 8, 2012 6/7
  16. 16. ExamplesGLB and LUB GLB LUB {x | 0 < x ≤ 3} {x | x2 − 3 < 0} x {y | y = x+1 , x ≥ 0} 3 {y | y = x2 , x = 0 Mathematics 4 () Completeness Axiom June 8, 2012 6/7
  17. 17. ExamplesGLB and LUB GLB LUB {x | 0 < x ≤ 3} {x | x2 − 3 < 0} x {y | y = x+1 , x ≥ 0} 3 {y | y = x2 , x = 0 3 {y | y = x2 , x ∈ Z, x = 0 Mathematics 4 () Completeness Axiom June 8, 2012 6/7
  18. 18. Homework 4Find the GLB and LUB of the following sets 3x 1 {x2 | x ∈ Z+ and − 3 ≤ 2 < 27} 2 { −2 | x ∈ Z and x = 0 and |x| < 5} x 3 {v | v ∈ Q and v 2 < 5} 4 {n | n = x26 , x ∈ R} +2 √ 5 { 9 − x2 | x ∈ R} Mathematics 4 () Completeness Axiom June 8, 2012 7/7

×