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Analysis > Michealmas > Flashcards

Michealmas Flashcards

1
Q

Define a statment.

A

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

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2
Q

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

A

When both A and B are true

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3
Q

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

A

When either A or B or both are true

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4
Q

When is the statment “not A” true?

A

When A is false

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5
Q

What does “not A” stand for?

A

The negation of A

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6
Q

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

A

If the outputs from the truth tables are the same

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7
Q

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

A
  1. ) Law of commutativity
  2. ) Law of associativity
  3. ) Law of distributivity
  4. ) De Morgan’s Law
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8
Q

What is the law of commutativity to do with statements?

A
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9
Q

What is the law of associativity to do with statements?

A
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10
Q

What is the law of distributivity to do with statements?

A
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11
Q

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

A
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12
Q

Define a set.

A

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

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13
Q

What does X ∪ Y stand for and mean?

A

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

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14
Q

What does X ∩ Y stand for and mean?

A

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

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15
Q

What does X \ Y stand for and mean?

A

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

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16
Q

What does Y ⊂ X stand for and mean?

A

Subset - The elements of Y are also elements of X

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17
Q

What does Yc for Y ⊂ X stand for and mean?

A

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

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18
Q

What is the law of commutativity to do with sets?

A
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19
Q

What is the law of associativity to do with sets?

A
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20
Q

What is the law of distributivity to do with sets?

A
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21
Q

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

A
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22
Q

How do you prove two sets are equal?

A

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

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23
Q

Give three main equivalence between the sets and statements.

A
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24
Q

Prove de morgan’s first law.

A
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25
Q

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

A
  1. ) Trichotomy
  2. ) Adition law
  3. ) Multiplication law
  4. ) Transitivity
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26
Q

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

A

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

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27
Q

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

A

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

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28
Q

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

A

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

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

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29
Q

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

A

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

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30
Q

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

A
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31
Q

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

A
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32
Q

Prove b.)

A

Ask Beth

33
Q

Define a real sequence.

A

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

34
Q

Define a limit of a sequence.

A
35
Q

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

A

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

36
Q

What is the ε interval around x*?

A
37
Q

What is uniqueness of the limit thereom (3.3)?

A

Every convergent sequence (zn)n∈ℕ has precisely one limit

38
Q

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

A
39
Q

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

A
40
Q

Finish this theorem, every convergent sequence (xn) is a… ?

A

bounded sequence

41
Q

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

A
42
Q

What is the squeezing theorem (3.5)?

A
43
Q

Prove the squeezing theorem.

A
44
Q

Finish this theorm (3.6) - Let yn → 0 as n → ∞. Let (xn) be a bounded sequence. Then we have xnyn

A

→ 0 as n → ∞

45
Q

Prove this theorem - Let yn → 0 as n → ∞. Let (xn) be a bounded sequence. Then we have xnyn → 0 as n → ∞.

A
46
Q

What is the calculus of limits theorem (3.7)?

A
47
Q

Prove calclulus of limits therorem.

A

Ask Beth

48
Q

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

A
  1. ) Exponentials beat powers
  2. ) Powers beat logs
49
Q

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

A

f(xn) → f(x*) as n → ∞

50
Q

Prove this collary.

A
51
Q

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

A

Convergent and we have

52
Q

Prove this proposition.

A
53
Q

What is the

A
54
Q

Define a limit of a complex sequence.

A
55
Q

What are the steps in a proof by induction?

A
  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
56
Q

How do you negate a statement?

A
57
Q

Negate the definition of a limit.

A
58
Q

When is a statement “if A then B” false?

A

If A is true but B is false

59
Q

What is the negation of “if A then B”?

A

A and (notB)

60
Q

What is the theorem about contrapositive statements?

A
61
Q

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

A

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

62
Q

Defne maximum/minimum.

A
63
Q

Define bounded above/below, upper/lower bound.

A
64
Q

Define supremum/infimum.

A
65
Q

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

A

The maximum

66
Q

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

A

Minimum

67
Q

What is the completeness axiom for ℝ?

A

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

68
Q

Define the image set of a function

A
69
Q

Prove part b of this proposition.

A
70
Q

Prove part a of this proposition.

A
71
Q

Define monotone increasing/decreasing.

A
72
Q

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

A
73
Q

Prove this thereom.

A
74
Q

Define a subsequence.

A
75
Q

Finish this proposition.

A
76
Q

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

A
77
Q

What is Bolzano - Weierstrass theorem?

A

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

78
Q

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

A

Analysis (2 decks)

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