Midterm Exam

Question 1

Closure under Addition

Answer 1

For every a, b € F, a + b € F

Question 2

Closure under Multiplication

Answer 2

For every a, b € F, a * b € F

Question 3

Associative Law of Addition

Answer 3

For every a, b, c € F, (a + b) + c = a + (b + c)

Question 4

Associative Law of Multiplication

Answer 4

For every a, b, c € F, (a * b) * c = a * (b * c)

Question 5

Commutative Law of Addition

Answer 5

For every a, b, € F, a + b = b + a

Question 6

Commutative Law of Multiplication

Answer 6

For every a, b € F, a * b = b * a

Question 7

Existence of Additive Identity

Answer 7

There exists an element 0 € F such that for every a € F, a + 0 = a

Question 8

Existence of Multiplicative Identity

Answer 8

There exists an element 1 € F such that 1 ≠ 0 and for every a € F, a * 1 = a

Question 9

Existence of Additive Inverse

Answer 9

For every a € F, there exist an element -a € F such that a + (-a) = 0

Question 10

Existence of Multiplicative Inverse

Answer 10

For every a € F with a ≠ 0, there exists an element a^-1 € F such that a * a^-1 = 1

Question 11

Distributive Law (good start to proofs when lost)

Answer 11

For every a, b, c € F a * (b + c) = (a * b) + (a * c)

Question 12

Ordered Field (all elements are comparable)

Answer 12

An ordered Field F with a relation <= that makes F into a totally ordered set such that:
1. x <= y implies x + z <= y + z (Addition preserves order)
2. x <= y and 0 <= z imply x * z <= y * z (Multiplication preserves order when multiplying by a positive number)

Question 13

Bounds - conceptually

Answer 13

Set = {1, 2} as a subset of Z+.
2, 3, 4 and 5 are all greater than or equal to every element of S.
They are all upper bounds of S that are in Z+.
2 is the least upper bound (supremum) of S
2 is the maximum of S

Question 14

Upper Bound

Answer 14

Let X be a totally ordered set and let S [be a subset or equal to the set of] X. An upper bound for the subset of S is an element B € X such that (“for every” s € S) s <= B. If the subset S has an upper bound, it is BOUNDED ABOVE.

Question 15

Supremum

Answer 15

An upper bound _B for S is a supremum if nothing smaller than _B is an upper bound. That is…
1. (“for every” s € S) s <= _B, i.e. _B is an upper bound
2. (“for every” y < _B)(“there exists an” a € S)
a > y, i.e. every element smaller than _B is not an upper bound
If sup S € S, it is called max S

Question 16

Lower Bound

Answer 16

Let X be a totally ordered set and let S [be subset or the set] X. A lower bound for the subset S is an element b € X such that (“for every” s € S) s >= b. If the subset S has a lower bound, then S is bounded below.

Question 17

Infimum (greatest lower bound)

Answer 17

A lower bound b (with dash on top) for S is an infimum if nothing larger than b (with a dash on top) is a lower bound. That is…
1. (“for every” s € S) s >= b (with a dash on top), i.e. b” is a lower bound
2. (“for every” y € b”) (“there exists an” a € S) a < y, i.e. every element larger than b” is not a lower bound.
If inf S € S, it is called the min S.

Question 18

Least Upper Bound Property

Answer 18

A totally ordered set X has the least upper bound property of every non-empty subset of X that is bounded above has a supremum.

Question 19

Greatest Lower Bound Property

Answer 19

A totally ordered set X has the greatest lower bound property of every non-empty subset of X that is bounded below has a infimum.

Question 20

Rationals: greatest lower bound property and least upper bound property

Answer 20

Rationals Q don’t have either property —> has ‘gaps’… these gaps are filled by the REAL NUMBERS R

Question 21

Completion

Answer 21

Every non-empty set in the Real numbers with an upper bound has a supremum.

Question 22

Triangle Inequality

Answer 22

|x + y| <= |x| + |y|