Real numbers T2

Question 1

What are the algebraic properties of the real numbers?

Answer 1

x + y = y + x - commutativity
(x+y)+z = x+(y+z) - associativity
there exists an identity element and an inverse

Question 2

what is the uniqueness of addiditive inverse

Answer 2

if we have a real number x and u and v are both additive inverses of x, u = v

Question 3

what are the order properties of the real numbers

Answer 3

trichotomy - for every x ∈ R, exactly one of the following holds:
x ∈ P , x = 0, −x ∈ P
compatibility - if x and y are in p, (x+y is also in p and so is xy)

Question 4

what is a supremum of a set

Answer 4

when x is an upper bound of S and for any y, if y is an upper bound of the set, then x is less than equal to y.

Question 5

if S is a subset of R, and has a supremum, it has exactly one supremum

Answer 5

if S is a subset of R, and has a supremum, it has exactly one supremum

Question 6

what is the Archimedean property

Answer 6

for any x in r, there exists n in the natural numbers such as x < n

Question 7

what is an infinum

Answer 7

let S be a subset of R that is bounded below. A number x is an infimum if x is a lower bound of S and for any y, if Y is a lower bound of S then y<=x