Completeness axiom

1. Completeness Axiom
Mathematics 4
June 8, 2012
Mathematics 4 () Completeness Axiom June 8, 2012 1 / 7

2. Denitions
Upper Bound and Lower Bound of a Set
Upper Bound of a Set
An upper bound(UB)of a set is a numbergreater than or equalto any
element of the setS.
Lower Bound of a Set
An lower bound(LB)of a set is a numberless than or equalto any
element of the setS.
Mathematics 4 () Completeness Axiom June 8, 2012 2 / 7

3. Denitions
Upper Bound and Lower Bound of a Set
Upper Bound of a Set
An upper bound(UB)of a set is a numbergreater than or equalto any
element of the setS.
Lower Bound of a Set
An lower bound(LB)of a set is a numberless than or equalto any
element of the setS.
Mathematics 4 () Completeness Axiom June 8, 2012 2 / 7

4. Denitions
Upper Bound and Lower Bound of a Set
Upper and Lower Bounds of a Set
Determine three upper bounds and three lower bounds for the following
sets, if these exists:
1
f1;2;3;4g
2
Z
+ 3
N f1;2;3g
Mathematics 4 () Completeness Axiom June 8, 2012 3 / 7

5. Denitions
Upper Bound and Lower Bound of a Set
Upper and Lower Bounds of a Set
Determine three upper bounds and three lower bounds for the following
sets, if these exists:
1
f1;2;3;4g
2
Z
+ 3
N f1;2;3g
Mathematics 4 () Completeness Axiom June 8, 2012 3 / 7

6. Denitions
Upper Bound and Lower Bound of a Set
Upper and Lower Bounds of a Set
Determine three upper bounds and three lower bounds for the following
sets, if these exists:
1
f1;2;3;4g
2
Z
+ 3
N f1;2;3g
Mathematics 4 () Completeness Axiom June 8, 2012 3 / 7

7. Denitions
Least Upper Bound and Greatest Lower Bound of a Set
Least Upper Bound of a Set
TheLeast Upper Bound(LUB) of a set is the smallest possible upper
bound.
Greatest Lower Bound of a Set
TheGreatest Lower Bound(GLB) of a set is the largest possible lower
bound.
Mathematics 4 () Completeness Axiom June 8, 2012 4 / 7

8. Denitions
Least Upper Bound and Greatest Lower Bound of a Set
Least Upper Bound of a Set
TheLeast Upper Bound(LUB) of a set is the smallest possible upper
bound.
Greatest Lower Bound of a Set
TheGreatest Lower Bound(GLB) of a set is the largest possible lower
bound.
Mathematics 4 () Completeness Axiom June 8, 2012 4 / 7

9. Examples
GLB and LUB
GLB LUB
f1;2;3;4g
Z
+
fxjx2R;jxj<4gN f1;2;3g
Mathematics 4 () Completeness Axiom June 8, 2012 5 / 7

10. Examples
GLB and LUB
GLB LUB
f1;2;3;4g
Z
+
fxjx2R;jxj<4gN f1;2;3g
Mathematics 4 () Completeness Axiom June 8, 2012 5 / 7

11. Examples
GLB and LUB
GLB LUB
f1;2;3;4g
Z
+
fxjx2R;jxj<4gN f1;2;3g
Mathematics 4 () Completeness Axiom June 8, 2012 5 / 7

12. Examples
GLB and LUB
GLB LUB
f1;2;3;4g
Z
+
fxjx2R;jxj<4gN f1;2;3g
Mathematics 4 () Completeness Axiom June 8, 2012 5 / 7

13. Examples
GLB and LUB
GLB LUB
fxj0< x3g
fxjx
2
3<0gfyjy=
x
x+1
;x0gfyjy=
3
x
2;x6= 0fyjy=
3
x
2;x2Z;x6= 0
Mathematics 4 () Completeness Axiom June 8, 2012 6 / 7

14. Examples
GLB and LUB
GLB LUB
fxj0< x3g
fxjx
2
3<0gfyjy=
x
x+1
;x0gfyjy=
3
x
2;x6= 0fyjy=
3
x
2;x2Z;x6= 0
Mathematics 4 () Completeness Axiom June 8, 2012 6 / 7

15. Examples
GLB and LUB
GLB LUB
fxj0< x3g
fxjx
2
3<0gfyjy=
x
x+1
;x0gfyjy=
3
x
2;x6= 0fyjy=
3
x
2;x2Z;x6= 0
Mathematics 4 () Completeness Axiom June 8, 2012 6 / 7

16. Examples
GLB and LUB
GLB LUB
fxj0< x3g
fxjx
2
3<0gfyjy=
x
x+1
;x0gfyjy=
3
x
2;x6= 0fyjy=
3
x
2;x2Z;x6= 0
Mathematics 4 () Completeness Axiom June 8, 2012 6 / 7

17. Examples
GLB and LUB
GLB LUB
fxj0< x3g
fxjx
2
3<0gfyjy=
x
x+1
;x0gfyjy=
3
x
2;x6= 0fyjy=
3
x
2;x2Z;x6= 0
Mathematics 4 () Completeness Axiom June 8, 2012 6 / 7

18. Homework 4
Find the GLB and LUB of the following sets
1
fx
2
jx2Z
+
and3
3x
2
<27g
2
f
2
x
jx2Zandx6= 0andjxj<5g
3
fvjv2Qandv
2
<5g
4
fnjn=
6
x
2
+2
;x2Rg
5
f
p
9x
2
jx2Rg
Mathematics 4 () Completeness Axiom June 8, 2012 7 / 7