Analysis 1 period 1.1

Question 1

Definition of subsets

Answer 1

Suppose that A and B are sets. A is a subset of B if for all x ∈ A it follows that x ∈ B.
A proper subset of B, denoted by A proper subset B if for all x is an element of A it follows that x ∈ B and there exists x ∈ B s.t x is not an element of A.

Question 2

What kind of operations do you have on sets?

Answer 2

• Intersection
• Union
• Complement
• Minus
• Disjoint

Question 3

Guideline for proofs with sets

Answer 3

1. If you want to proff a = B, proof that a is a proper subset of B and otherwise
2. If you want to proof a statement with for all, then take an arbitrary situation and proof the statement for that situation
3. If you want to proof a statement with there exists, try to find a specific situation that fits the statement
4. If you want to proof an ‘If then’ statement you need to assume the if statement and proof the then statement
5. If you want to proof an if and only if statement you need to proof both ways.

Question 4

What is a relation?

Answer 4

A relation h is a set of ordered pairs(x,y) where x ∈ A and y ∈ B. This relations links two sets to each other. h: A -> B
The visualization is called the graph of a relation, also called a curve. Two relations are equal if they have the same graph.

Question 5

What is a function?

Answer 5

A relation f is a function if it doesn’t contain two different ordered pairs with the same first coordinate. In math, f is a function if and only if: for all (x1,y1) (x2,y2) ∈ f with x1 = x2 it follows that y1 = y2

A relation f is a function when you can say that every x value has an unique y value

Question 6

Domain

An other word for domain is preimage

Answer 6

The domain/preimage of a function f, denoted by Df is defined by Df = {x|(x,y) ∈ f}

Question 7

Range

Other word for range is image

Answer 7

Denoted by Rf is defined by: Rf = {y|(x,y)∈ f and x ∈ Df}

Question 8

How do you find the maximum domain of f?

There are 3 functions you have to take into account

Answer 8

1. For n^√g(x) with n is even, it must hold that g(x) >= 0
2. For log a (g(x)) is must hold that g(x) > 0
3. For f(x) / g(x) it must hold that g(x) =/ 0

=/ -> not equal , >= -> equal or bigger than

Question 9

Injective

Answer 9

If ∀x1,x2 ∈A; f(x1)=f(x2) it follows that x1 = x2

Question 10

Surjective

Answer 10

f is surjective If ∀ y∈B ∃x∈A: f(x)=y. So in case of surjection Rf = B

Question 11

Bijection

Answer 11

When f is both injective and surjective

you have to proof both the ways in order to conclude that f is bijective

Question 12

How do you proof injection?

Answer 12

1. x1,x2 ∈A; f(x1)=f(x2) be given
2. Use algebraic rules to work to the conclusion x1=x2
3. Conclude that f is injective

If you can’t reach this conclusion, you must give an example where two different x values give the same y value. This example concludes that f is not injective

Question 13

How do you prove surjection?

Answer 13

1. Let y∈B be given
2. Use algebraic rules isolate x and conclude x∈A
3. Conclude that f is surjective

If you can’t reach this conclusion, you must give a example where a number y∈B cannot be reached with f with domain Df. This example concludes that f is not surjective

Question 14

When is a function even and when is and function odd?

Answer 14

• f is an even function if and only if f(-x) = f(x) for all x ∈ A.
• f is an odd function if and only if f(-x) = -f(x) for all x ∈ A

Question 15

Proof even

Answer 15

1. Let x ∈ A be given
2. Define f(-x) and use algebra to prove that f(-x) = f(x)

If f is neither, then give an example of a number x ∈ A which gives f(-x) =/ f(x) and f(-x) =/ -f(x)

Question 16

Proof odd

Answer 16

1. Let x ∈ A be given
2. Define f(-x) and use algebra to prove that f(-x) = -f(x)

If f is neither, then give an example of a number x ∈ A which gives f(-x) =/ f(x) and f(-x) =/ -f(x)

Question 17

Monotone functions

Answer 17

• f is increasing<=> f(x1) <= f(x2) for all x1,x2 ∈ A; x1<x2
• f is strictly increasing <=> f(x1) < f(x2) for all x1,x2 ∈A; x1<x2
• f is decreasing <=> f(x1) >= f(x2) for all x1,x2 ∈A; x1<x2
• f is strictly decreasing <=> f(x1) > f(x2) for all x1,x2 ∈ A; x1<x2
• f is constant <=> f(x1) = f(x2) for all x1,x2 ∈A

If a function is increasing or decreasing then it is monotone. Same counts for strictly monotone

Question 18

Proving monotone functions

There are three steps

Answer 18

1. use a draft: start with f(x1) < f(x2)
2. Use algebra to get a statement about x1 and x2
   * x1 < x2 f is strictly increasing
   * x1 > x2 f is strictly decreasing
   * if you can’t conclude any of these options, f is none of them
3. Now we can start with the real proof:
   1.1 In case of non constant: Start with the expression you found about x1 and x2 and work in the opposite direction until you reached f(x1) < f(x2) and draw your conclusion
   1.2 In case of a constant function: Choose three coordinates (x1,y1), (x2,y2) and (x3,y3) where the function goes up, down and up again or down up and down again

Question 19

Absolute value of any real number

Answer 19

Denote by |x| and is defined by
|x| = { x if x >= 0
-x if x < 0
The absolute value function of f(x) = |x| with x ∈ R

Question 20

Bounded functions

Give the options

Answer 20

consider the function f: A -> B
* f is bounded below <=> there exists M ∈ R: f(x) >= M for all x ∈ A
* f is bounded above <=> there exists M ∈ R; f(x) <= M for all x ∈ A
* f is bounded <=> there exists M ∈ R+; |f(x)| <= M for all x ∈ A

f is bounded <=> there exists M∈R+; -M <= f(x) <= M for all x ∈ A
So f is bounded <=> f is bounded below and bounded above

Question 21

supremum and infimum versus maximum and minimum

Answer 21

Consider a bounded function f : a-> B
* If the global maximum of f exists, then sup f = max f
* If the global minimum of f exists, then inf f = min f

Question 22

Composition function

Suppose functions f : A->R and g : B = f(A) -> B. The composition function of g on f, g ○ f

Answer 22

g○f, maps set A into R.
In math: g ○ f : A->R
The function g ○ f is defined by (g○f)(x) = g(f(x))

Question 23

Mathematical induction

Answer 23

if p(n) is a statement for all n∈N s.t
a. p(1) is true, and
b. for each m∈N, if p(m) is true, then p(m+1) is true

then p(n) is true for all n∈N

Question 24

General mathematical induction

Answer 24

if p(n) is a statement for all n>=k for k∈N s.t.
a. p(k) is true, and
b. or each natural number m >=k, if p(m) is true, then p(m+1) is true

then p(n) is true for all n >= k

Question 25

Procedure mathematical induction:

Answer 25

1. Basis of the induction: Show that p(k) is true
2. Induction hypothesis (IH): Assume that for a certain m>= k that p(m) is true
3. Inductive step: Prove that p(m+1) is true

Question 26

Binomial coefficient

Answer 26

(n k) n! / (k! * (n-k)!)
where k and n are integers s.t
0 <= k <= n

Question 27

Strong induction

Answer 27

If p(n) is a statement for all n ∈N and m ∈ N s.t.
a. p(1), p(2),….,p(m) are true
b. For all k>=m if p(i) is true for i ∈ {1,…..,k} then p(k+1) is true
then p(n) is true for all n ∈ N

Question 28

Procedure strong induction

Answer 28

1. Basis of the induction: Show that p(n) is true for all n<=m
2. Induction hypothesis (IH): Assume for a certain k>=m s.t. p(i) is true for i∈{1,…..,k}
3. Inductive step: Prove that p(k+1) is true

Question 29

Inverse function

Answer 29

A function f is invertible if and only if its inverse relation is a function. f^(-1) denotes the inverse of f.

A function f is invertible if and only if f is one-to-one. That is, if f is injective

Question 30

Inverse relations

Answer 30

The inverse of a relation S is the relation T, where T = {(y,x)|(x,y)∈S}. The graph of T is obtained from the graph of S by reflecting the graph of S along the diagonal line y=x

Question 31

Procedure finding the inverse function

Answer 31

If we have an injective function f and we wish to find its inverse, we perform three steps:
1. replace f(x) by y
2. Interchange the variables x and y
3. Solve for y and label it f^(-1)(x)

Question 32

Definition: Image

Suppose the function f : A -> B

Answer 32

The image of the function f is defined by
f(C) = {f(x)∈B : x∈C}

Question 33

Procedure: Image

Answer 33

1. Find the inverse relation f^(-1) and write it in terms of y so f^(-1)(y)
2. Solve the equation or inequalities in terms of y
3. Put your answer in set notation for the image

Question 34

Definition: preimage

Suppose the function f : A -> B

Answer 34

The preimage of the function f is defined by
f^(-1)(C) = { x∈A : f(x) ∈ C}

Question 35

Procedure: preimage

Answer 35

1. Solve the equation or inequalities in the terms of y
2. Put your answer in set notation for the preimage

Question 36

Inverse of bijections

Answer 36

1. Suppose that f : A -> B is a bijection; then
   1.1 (f-1 ○ f )(x) = x for all x ∈ A
   1.2 (f ○ f-1)(x) = y for all y ∈ B
2. The function f : A -> B is a bijection if and only if f-1 : B -> A is a bijection

Question 37

What are the triangle equalities

Answer 37

1. |a + b| <= |a| + |b|
2. |a - b| >= |a| - |b|
3. ||a| - |b|| <= |a - b|