The Rational and Real Numbers

Question 1

For x∈Q how do we denote the absolute value of x?

Answer 1

|x|={x, if x≥ 0

{−x, if x ≤ 0

Question 2

When is a set A ⊆ Q bounded above by α ∈ Q?

Answer 2

If for all x ∈ A we have x ≤ α.

Question 3

When is a set A ⊆ Q bounded below by α ∈ Q?

Answer 3

If for all x ∈ A we have x ≥ α.

Question 4

What is meant if a set A ⊆ Q is bounded?

Answer 4

If A is bounded above and below we say that A is bounded.

Alternatively if there exists α ∈Q such that for every x ∈ A we have that |x|≤ α.

Question 5

Is √2 is rational, true or flase?

Answer 5

false, √2 is not rational.

Question 6

Define the completeness axiom.

Answer 6

Every non-empty subset A of R which is bounded above must have a least upper bound.

Question 7

Define closure under addition.

Answer 7

For all x, y ∈ Q we have that x + y ∈ Q

Question 8

Define associativity under addition.

Answer 8

For all x, y, z ∈ Q we have that x + (y +

z) = (x + y) + z.

Question 9

Show zero is the additive identity

Answer 9

For all x ∈ Q we have x + 0 = x.

Question 10

Define how every element has an additive inverse.

Answer 10

For all x ∈ Q we have −x ∈ Q and x + (−x) = 0.

Question 11

Define what is meant by addition is commutative.

Answer 11

For all x, y ∈ Q we have x + y = y + x.

Question 12

Define closure under multiplication.

Answer 12

For all x, y ∈ Q we have xy ∈ Q.

Question 13

Define associativity under multiplication.

Answer 13

For all x, y, z ∈ Q we have (xy)z = x(yz).

Question 14

Show one is the multiplicative identity.

Answer 14

For all x ∈ Q we have 1 · x = x.

Question 15

Show all nonzero rationals have a multiplicative inverse.

Answer 15

For all x ∈ Q where x≠0 we have that x^−1 = 1/x ∈ Q and x^−1 · x = 1.

Question 16

Define what is meant by multiplication is commutative

Answer 16

For all x, y ∈ Q we have that xy = yx.

Question 17

Define distributivity law.

Answer 17

For all x, y, z ∈ Q we have x(y + z) = xy + xz

Question 18

Define transitivity.

Answer 18

For all x, y, z ∈ Q if x < y and y < z then x < z.

Question 19

Define trichotomy.

Answer 19

For all x, y ∈ Q either x < y, x = y or y < x.

Question 20

Define compatibility with addition

Answer 20

For all x, y, z ∈ Q if x < y then x + z < y + z.

Question 21

Define compatibility with multiplication.

Answer 21

For all x, y, z ∈ Q if x < y and z > 0 then zx < zy.

Question 22

Properties of x∈R.

Answer 22

|x|≥ 0 with |x| = 0 if and only if x = 0
|xy| = |x||y|
|x^2| = |x|^2 = x^2

Question 23

Define the triangle inequality.

Answer 23

For all x,y∈R we have

|x + y|≤|x|+|y|

Question 24

For a,b,c ∈R we have that a^2 ≥ …. ?

Answer 24

0

Question 25

For a,b,c ∈R we have that if a,b ≥ 0 then …. ?

Answer 25

a ≥ b if and only if a^2 ≥ b^2

Question 26

For a,b,c ∈R we have that if a > 0 then …. ?

Answer 26

4ac−b^2 ≥ 0 if and only if for all x∈R, ax^2 + bx + c ≥ 0.

Question 27

For all n∈N we have that 2^n > n, true or false?

Answer 27

true, 2^n > n