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# Completeness axiom

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### 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