Michealmas Flashcards
Define a statment.
A statement is a sentance which is either true or flase but not both
When can the statement “A and B” be true?
When both A and B are true
When can the statement “A or B” be ture?
When either A or B or both are true
When is the statment “not A” true?
When A is false
What does “not A” stand for?
The negation of A
How do truth tables allow us to say if two statments are the equivalent?
If the outputs from the truth tables are the same
What are the name of the four laws to do with sets and statments?
- ) Law of commutativity
- ) Law of associativity
- ) Law of distributivity
- ) De Morgan’s Law
What is the law of commutativity to do with statements?
What is the law of associativity to do with statements?
What is the law of distributivity to do with statements?
What is de morgan’s law to do with statements?
Define a set.
A set is a well defined unordered collection of elements where each element is contained only once
What does X ∪ Y stand for and mean?
Union - The set of all elements in at least one of X and Y
What does X ∩ Y stand for and mean?
Intersection - The set of all elements contained in both X and Y
What does X \ Y stand for and mean?
Difference - The set of all elements contained in X but not in Y
What does Y ⊂ X stand for and mean?
Subset - The elements of Y are also elements of X
What does Yc for Y ⊂ X stand for and mean?
Compliment - The set of elements of X which are not in Y
What is the law of commutativity to do with sets?
What is the law of associativity to do with sets?
What is the law of distributivity to do with sets?
What is de morgan’s law to do with sets?
How do you prove two sets are equal?
We prove every element of one set is also an element of the other set
Give three main equivalence between the sets and statements.
Prove de morgan’s first law.
What four properties to the order realtions: <, >, ≤, ≥ satisfy?
- ) Trichotomy
- ) Adition law
- ) Multiplication law
- ) Transitivity
What is the trichotomy property that the order relations: <, >, ≥, ≤ satisfy?
Either x < y, x=y or x >y
What is the addition law that the order relations: <, >, ≥, ≤ satisfy?
If x < y and a ∈ ℝ, then x + a < y + a
What is the multiplciation law that the order relations: <, >, ≥, ≤ satisfy?
If x < y and c > 0, then cx < cy
If x > y and c < 0, then cx > cy
What is the transitivity property that the order relations: <, >, ≥, ≤ satisfy?
If x < y and y < z, then x < z
Prove that if x < y and a < b then x + a < y + b.
What four rules are important when considering inequalities involing absolute values?
Prove b.)
Ask Beth
Define a real sequence.
A real sequence is a function from ℕ to ℝ, we denote such sequecne by (xn)n∈ℕ.
Define a limit of a sequence.
What is roughly the difference between a convergent and divergent sequence?
Convergnet sequences have a limit where as divergent ones don’t
What is the ε interval around x*?
What is uniqueness of the limit thereom (3.3)?
Every convergent sequence (zn)n∈ℕ has precisely one limit
Prove the uniqueness of the limit therorem - Every convergent sequence (zn)n∈ℕ has precisely one limit
What is the theorem about what a boudned sequence is (3.4) ?
Finish this theorem, every convergent sequence (xn) is a… ?
bounded sequence
Prove every convergent sequence (xn) is a bounded sequence.
What is the squeezing theorem (3.5)?
Prove the squeezing theorem.
Finish this theorm (3.6) - Let yn → 0 as n → ∞. Let (xn) be a bounded sequence. Then we have xnyn
→ 0 as n → ∞
Prove this theorem - Let yn → 0 as n → ∞. Let (xn) be a bounded sequence. Then we have xnyn → 0 as n → ∞.
What is the calculus of limits theorem (3.7)?
Prove calclulus of limits therorem.
Ask Beth
When determining limits of functions what four contrinuous functions can we say beat each other?
- ) Exponentials beat powers
- ) Powers beat logs
Finish this theorem (3.9) - If xn → x* as n → ∞ and f is continuous at x*, we have
f(xn) → f(x*) as n → ∞
Prove this collary.
Finish this theorem - Let |c| < 1. Then the sequence (cn) is… ?
Convergent and we have
Prove this proposition.
What is the
Define a limit of a complex sequence.
What are the steps in a proof by induction?
- ) Start and prove for A(n0)
- ) Assume A(k) is true for some k ≥ n0
- ) Derive that A(k+1) is true
- ) Then A(n0) true ⟹ A(n0 + 1) true ⟹ A(n0 + 2) true
- ) Therefore A(n) is true ∀ n ≥ n0 by induction
How do you negate a statement?
Negate the definition of a limit.
When is a statement “if A then B” false?
If A is true but B is false
What is the negation of “if A then B”?
A and (notB)
What is the theorem about contrapositive statements?
What is the difference between an indirect proof using negation and a direct contrapositive proof?
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
Defne maximum/minimum.
Define bounded above/below, upper/lower bound.
Define supremum/infimum.
Does the supremum or maximum have to be part of the set?
The maximum
Dies the minimum or infinmum has to be part of the set?
Minimum
What is the completeness axiom for ℝ?
Every non-empty set of real numbers which is bounded has a supremum
Define the image set of a function
Prove part b of this proposition.
Prove part a of this proposition.
Define monotone increasing/decreasing.
Finish this theroem - Let (xn) be a monotone increasinf real sequence. If (xn) is bounded, then … ?
Prove this thereom.
Define a subsequence.
Finish this proposition.
Prove this lemma - Every real sequecnce (xn) contains a subsequence which is either increasing or decreasing.
What is Bolzano - Weierstrass theorem?
Let (xn) be a bounded real sequence. Then (xn) has a subsequence which is convergent.
Prove Bolzano-Weierstrass theorem - Let (xn) be a bounded real sequence. Then (xn) has a subsequence which is convergent.
