Michealmas

Question 1

Define a statment.

Answer 1

A statement is a sentance which is either true or flase but not both

Question 2

When can the statement “A and B” be true?

Answer 2

When both A and B are true

Question 3

When can the statement “A or B” be ture?

Answer 3

When either A or B or both are true

Question 4

When is the statment “not A” true?

Answer 4

When A is false

Question 5

What does “not A” stand for?

Answer 5

The negation of A

Question 6

How do truth tables allow us to say if two statments are the equivalent?

Answer 6

If the outputs from the truth tables are the same

Question 7

What are the name of the four laws to do with sets and statments?

Answer 7

1. ) Law of commutativity
2. ) Law of associativity
3. ) Law of distributivity
4. ) De Morgan’s Law

Question 8

What is the law of commutativity to do with statements?

Answer 8

Question 9

What is the law of associativity to do with statements?

Answer 9

Question 10

What is the law of distributivity to do with statements?

Answer 10

Question 11

What is de morgan’s law to do with statements?

Answer 11

Question 12

Define a set.

Answer 12

A set is a well defined unordered collection of elements where each element is contained only once

Question 13

What does X ∪ Y stand for and mean?

Answer 13

Union - The set of all elements in at least one of X and Y

Question 14

What does X ∩ Y stand for and mean?

Answer 14

Intersection - The set of all elements contained in both X and Y

Question 15

What does X \ Y stand for and mean?

Answer 15

Difference - The set of all elements contained in X but not in Y

Question 16

What does Y ⊂ X stand for and mean?

Answer 16

Subset - The elements of Y are also elements of X

Question 17

What does Y^c for Y ⊂ X stand for and mean?

Answer 17

Compliment - The set of elements of X which are not in Y

Question 18

What is the law of commutativity to do with sets?

Answer 18

Question 19

What is the law of associativity to do with sets?

Answer 19

Question 20

What is the law of distributivity to do with sets?

Answer 20

Question 21

What is de morgan’s law to do with sets?

Answer 21

Question 22

How do you prove two sets are equal?

Answer 22

We prove every element of one set is also an element of the other set

Question 23

Give three main equivalence between the sets and statements.

Answer 23

Question 24

Prove de morgan’s first law.

Answer 24

Question 25

What four properties to the order realtions: \<, \>, ≤, ≥ satisfy?

Answer 25

1. ) Trichotomy 2. ) Adition law 3. ) Multiplication law 3. ) Transitivity

Question 26

What is the trichotomy property that the order relations: \<, \>, ≥, ≤ satisfy?

Answer 26

Either x \< y, x=y or x \>y

Question 27

What is the addition law that the order relations: \<, \>, ≥, ≤ satisfy?

Answer 27

If x \< y and a ∈ ℝ, then x + a \< y + a

Question 28

What is the multiplciation law that the order relations: \<, \>, ≥, ≤ satisfy?

Answer 28

If x \< y and c \> 0, then cx \< cy If x \> y and c \< 0, then cx \> cy

Question 29

What is the transitivity property that the order relations: \<, \>, ≥, ≤ satisfy?

Answer 29

If x \< y and y \< z, then x \< z

Question 30

Prove that if x \< y and a \< b then x + a \< y + b.

Answer 30

Question 31

What four rules are important when considering inequalities involing absolute values?

Answer 31

Question 32

Prove b.)

Answer 32

Ask Beth

Question 33

Define a real sequence.

Answer 33

A real sequence is a function from ℕ to ℝ, we denote such sequecne by (x_n)_n∈ℕ.

Question 34

Define a limit of a sequence.

Answer 34

Question 35

What is roughly the difference between a convergent and divergent sequence?

Answer 35

Convergnet sequences have a limit where as divergent ones don't

Question 36

What is the ε interval around x\*?

Answer 36

Question 37

What is uniqueness of the limit thereom (3.3)?

Answer 37

Every convergent sequence (z_n)_n∈ℕ has precisely one limit

Question 38

Prove the uniqueness of the limit therorem - Every convergent sequence (zn)n∈ℕ has precisely one limit

Answer 38

Question 39

What is the theorem about what a boudned sequence is (3.4) ?

Answer 39

Question 40

Finish this theorem, every convergent sequence (x_n) is a... ?

Answer 40

bounded sequence

Question 41

Prove every convergent sequence (x_n) is a bounded sequence.

Answer 41

Question 42

What is the squeezing theorem (3.5)?

Answer 42

Question 43

Prove the squeezing theorem.

Answer 43

Question 44

Finish this theorm (3.6) - Let y_n → 0 as n → ∞. Let (x_n) be a bounded sequence. Then we have x_ny_n

Answer 44

→ 0 as n → ∞

Question 45

Prove this theorem - Let y_n → 0 as n → ∞. Let (x_n) be a bounded sequence. Then we have x_ny_n → 0 as n → ∞.

Answer 45

Question 46

What is the calculus of limits theorem (3.7)?

Answer 46

Question 47

Prove calclulus of limits therorem.

Answer 47

Ask Beth

Question 48

When determining limits of functions what four contrinuous functions can we say beat each other?

Answer 48

1. ) Exponentials beat powers 2. ) Powers beat logs

Question 49

Finish this theorem (3.9) - If x_n → x\* as n → ∞ and f is continuous at x\*, we have

Answer 49

f(x_n) → f(x\*) as n → ∞

Question 50

Prove this collary.

Answer 50

Question 51

Finish this theorem - Let |c| \< 1. Then the sequence (cⁿ) is... ?

Answer 51

Convergent and we have

Question 52

Prove this proposition.

Answer 52

Question 53

What is the

Answer 53

Question 54

Define a limit of a complex sequence.

Answer 54

Question 55

What are the steps in a proof by induction?

Answer 55

1. ) Start and prove for A(n₀) 2. ) Assume A(k) is true for some k ≥ n₀ 3. ) Derive that A(k+1) is true 4. ) Then A(n₀) true ⟹ A(n₀ + 1) true ⟹ A(n₀ + 2) true 5. ) Therefore A(n) is true ∀ n ≥ n₀ by induction

Question 56

How do you negate a statement?

Answer 56

Question 57

Negate the definition of a limit.

Answer 57

Question 58

When is a statement "if A then B" false?

Answer 58

If A is true but B is false

Question 59

What is the negation of "if A then B"?

Answer 59

A and (notB)

Question 60

What is the theorem about contrapositive statements?

Answer 60

Question 61

What is the difference between an indirect proof using negation and a direct contrapositive proof?

Answer 61

For an indirect proof using negation we start with the negation of the original statement adn derive a contraction where as the contrapositive proof we start with the contrapositive statement and prove it directly

Question 62

Defne maximum/minimum.

Answer 62

Question 63

Define bounded above/below, upper/lower bound.

Answer 63

Question 64

Define supremum/infimum.

Answer 64

Question 65

Does the supremum or maximum have to be part of the set?

Answer 65

The maximum

Question 66

Dies the minimum or infinmum has to be part of the set?

Answer 66

Minimum

Question 67

What is the completeness axiom for ℝ?

Answer 67

Every non-empty set of real numbers which is bounded has a supremum

Question 68

Define the image set of a function

Answer 68

Question 69

Prove part b of this proposition.

Answer 69

Question 70

Prove part a of this proposition.

Answer 70

Question 71

Define monotone increasing/decreasing.

Answer 71

Question 72

Finish this theroem - Let (x_n) be a monotone increasinf real sequence. If (x_n) is bounded, then ... ?

Answer 72

Question 73

Prove this thereom.

Answer 73

Question 74

Define a subsequence.

Answer 74

Question 75

Finish this proposition.

Answer 75

Question 76

Prove this lemma - Every real sequecnce (x_n) contains a subsequence which is either increasing or decreasing.

Answer 76

Question 77

What is Bolzano - Weierstrass theorem?

Answer 77

Let (x_n) be a bounded real sequence. Then (x_n) has a subsequence which is convergent.

Question 78

Prove Bolzano-Weierstrass theorem - Let (xn) be a bounded real sequence. Then (xn) has a subsequence which is convergent.

Answer 78