Analysis

Question 1

What are the properties of real numbers (7)

Answer 1

• For every x, y ∈ R, there are numbers x + y ∈ R (sum) and xy ∈ R (product)
• Sum and product are commutative: for any x, y ∈R,
  x + y = y + x and xy = yx.
• Sum and product are associative: for any x, y, z E R,
  (x + y) + z = x + (y + z) and (xy)z = x(yz).
• Identities:
  x ∈ R, x + 0 = x and x1 = x.
• Additive inverse: For every x ∈ R, there is a (-x) ∈ R with
  x + (-x) =0
• Multiplicative inverse: For every x E R with x does not equal 0, there is a x^-1 ∈ R with xx^1 = 1.
• Distributivity: For x, y, z ∈ R, we have x (y + z) = xy + xz.

Question 2

What does x>y mean

Answer 2

x-y>0

Question 3

What are the properties of modulus (6)

Answer 3

For any x∈R, |x| = max (x,-x)
* For any x∈R and a ≥ 0, we have |x| ≤ a if and only if
x ≤ a and -x ≤ a.
* For any x ∈ R, we have
|x| ≥ 0, with |x| = 0 if and only if x＝0.
* For any x∈R, - x ≤ x ≤ x .
* For a, b ∈ R, |ab| = |a||b|
* For a, b ∈ R,
|a + b| ≤ |a|+|b| (triangle inequality)
* For a, b∈R,
|a - b| ≥ ||a| - |b|| (reverse triangle inequality)

Question 4

Define bounded above in terms of a set A ⊆ R

Answer 4

• for every x ⊆ A, x ≤ M
• M is classed as an upper bound
• the reverse is true for bounded below

Question 5

What is the definition of a maximum

Answer 5

for m ∈ A, if x ≤ m for all x∈A then m is the maximum

Question 6

What is the definition of the supremum

Answer 6

The smallest number that is greater than or equal to every element in a set

Question 7

A⊆R, then supA = s if and only if: (2)

Answer 7

• if x ∈ A, then x