The document defines upper and lower bounds of a set as numbers greater than or equal to and less than or equal to elements of the set, respectively. It also defines the least upper bound and greatest lower bound as the smallest upper bound and largest lower bound. Several examples of sets are given with their greatest lower and least upper bounds identified. The homework assigns finding the greatest lower and least upper bounds of additional sets.

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

2. Definitions
Upper Bound and Lower Bound of a Set
Upper Bound of a Set
An upper bound (UB) of a set is a number greater than or equal to any
element of the set S.
Mathematics 4 () Completeness Axiom June 8, 2012 2/7

3. Definitions
Upper Bound and Lower Bound of a Set
Upper Bound of a Set
An upper bound (UB) of a set is a number greater than or equal to any
element of the set S.
Lower Bound of a Set
An lower bound (LB) of a set is a number less than or equal to any
element of the set S.
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4. Definitions
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 {1, 2, 3, 4}
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5. Definitions
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 {1, 2, 3, 4}
2 Z+
Mathematics 4 () Completeness Axiom June 8, 2012 3/7

6. Definitions
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 {1, 2, 3, 4}
2 Z+
3 N − {1, 2, 3}
Mathematics 4 () Completeness Axiom June 8, 2012 3/7

7. Definitions
Least Upper Bound and Greatest Lower Bound of a Set
Least Upper Bound of a Set
The Least Upper Bound (LUB) of a set is the smallest possible upper
bound.
Mathematics 4 () Completeness Axiom June 8, 2012 4/7

8. Definitions
Least Upper Bound and Greatest Lower Bound of a Set
Least Upper Bound of a Set
The Least Upper Bound (LUB) of a set is the smallest possible upper
bound.
Greatest Lower Bound of a Set
The Greatest Lower Bound (GLB) of a set is the largest possible lower
bound.
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9. Examples
GLB and LUB
GLB LUB
{1, 2, 3, 4}
Mathematics 4 () Completeness Axiom June 8, 2012 5/7

10. Examples
GLB and LUB
GLB LUB
{1, 2, 3, 4}
Z+
Mathematics 4 () Completeness Axiom June 8, 2012 5/7

11. Examples
GLB and LUB
GLB LUB
{1, 2, 3, 4}
Z+
{x | x ∈ R, |x| < 4}
Mathematics 4 () Completeness Axiom June 8, 2012 5/7

12. Examples
GLB 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. Examples
GLB and LUB
GLB LUB
{x | 0 < x ≤ 3}
Mathematics 4 () Completeness Axiom June 8, 2012 6/7

14. Examples
GLB and LUB
GLB LUB
{x | 0 < x ≤ 3}
{x | x2 − 3 < 0}
Mathematics 4 () Completeness Axiom June 8, 2012 6/7

15. Examples
GLB 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. Examples
GLB 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. Examples
GLB 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. Homework 4
Find 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}
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