Numbers and functions

Question 1

cardinality

Answer 1

number of elements in a set

Question 2

Injective

Answer 2

A function is said to be injective if f(x)=f(y) ⇒ x=y

Question 3

Surjective

Answer 3

A function f:A->B is surjective if f(A)=B. Each element of the codomain is mapped to by some element of the domain.

Question 4

Bijective

Answer 4

Both injective and Surjective

Question 5

Contrapositive

Answer 5

A⇒B, the contrapositive is: not B⇒not A (always true if A⇒B is true)

Question 6

Converse

Answer 6

A⇒B, the converse is B⇒A (not always true)

Question 7

How do you prove two sets are equal?

Answer 7

Prove the lhs is a member of the rhs and prove that the rhs is a member of the lhs.

Question 8

Bounded above

Answer 8

∃M∈ℝ, ∀x∈S, such that x≤M

Question 9

Bounded Below

Answer 9

∃M∈ℝ, ∀x∈S, such that x≥M

Question 10

Unbounded above

Answer 10

∀M∈ℝ, ∃x∈S, such that x>M

Question 11

Maximum

Answer 11

Let S be bounded above and suppose that there exists an upper bound M of S such that M∈S. M is the maximum of S.

Question 12

infimum

Answer 12

Greatest lower bound

Question 13

supremum

Answer 13

Least upper bound

Question 14

minimum

Answer 14

Let S be bounded below and suppose that there exists a lower bound m of s such that m∈S.

Question 15

Axiom of completeness

Answer 15

Every nonempty set of real numbers which is bounded above has a supremum. Every non-empty set of Real numbers which is bounded below has an infimum.

Question 16

Bounded

Answer 16

|S(n)|≤M, ∀n∈ℕ

Question 17

Convergence

Answer 17

∀ℇ>0, ∃n(0)∈ℕ such that ∀n≥n(0), we have |S(n)-𝓁|

Question 18

Divergence

Answer 18

If the sequence S(n) does not converge to any limit it is said to diverge.

Question 19

Divergence to ∞

Answer 19

The sequence S(n) is said to diverge to ∞, for which we write S(n)→∞, if for every positive real number H, there exists n(0) such that ∀n≥n(0) we have S(n)>H.

Question 20

Triangle inequality

Answer 20

|x+y| ≤ |x|+|y|

Question 21

Sandwich Theorem

Answer 21

Let r(n)→𝓁 and t(n)→𝓁 as n→∞ and suppose that r(n)≤s(n)≤t(n) for all n∈ℕ. Then s(n)→𝓁 as n→∞.

Question 22

If s(n)→0 as n→∞ and t(n) is a bounded sequence, then s(n)t(n)→? as n→∞

Answer 22

s(n)t(n)→0

Question 23

If s(n)→𝓁 and t(n)→∞ as n→∞, then s(n)t(n)→? as n→∞

Answer 23

s(n)t(n)→∞

Question 24

Let s(n)→∞ or s(n)→-∞ as n→∞ and s(n)≠0 for all n. then 1/s(n)→? as n→∞

Answer 24

1/s(n)→0

Question 25

Put these functions in order of their rate of convergence:

Exponentials, logarithms, factorials, powers

Answer 25

Slowest: Logarithms
Powers
Exponentials
Fastest: Factorials

Question 26

Bernoulli inequality

Answer 26

For every k∈ℕ and x≥-1, one has (1+x)^k≥1+kx

Question 27

Little o

Answer 27

Let s(n) and t(n) be two sequences, such that t(n)≠0 for all n. Suppose that as n→∞, s(n)/t(n)→0. Then we can say s(n)=o(t(n)) as n→∞

Question 28

Big O

Answer 28

Let s(n) and t(n) be sequences. Suppose that there exists C>0 such that for all n∈ℕ, one has |s(n)|≤C|t(n)|. Then one writes s(n)=Ot(n) as n→∞. I.e. s(n)/t(n) is Bounded.

Question 29

lim(1+1/n)^n

Answer 29

e

Question 30

increasing sequence

Answer 30

s(n+1)>s(n) for all n∈ℕ

Question 31

decreasing sequence

Answer 31

s(n+1)

Question 32

non-decreasing sequence

Answer 32

s(n+1)≥s(n) for all n∈ℕ

Question 33

non-increasing sequence

Answer 33

s(n+1)≤s(n) for all n∈ℕ

Question 34

monotone sequence

Answer 34

neither non-increasing nor non-decreasing

Question 35

True or false:

Every non-increasing sequence which is bounded above is convergent.

Answer 35

FALSE

Every non-decreasing sequence which is bounded above is convergent.

Question 36

The Bolzano-Weierstrass Theorem

Answer 36

Every bounded sequence has a convergent subsequence, which converges to a limit 𝓵 which is between the bounds of the original sequence.

Question 37

Cauchy Sequence

Answer 37

s(n) is a cauchy sequence if for all ℇ>0 there exists a n(0) such that |s(m)-s(n)|

Question 38

Absolute convergence

Answer 38

if the series 𝚺|a(k)| from k=1 to ∞ is convergent then 𝚺a(k) from k=1 to ∞ is absolutely convergent

Question 39

conditional convergence

Answer 39

If the series 𝚺a(k) from k=1 to ∞ converges but 𝚺|a(k)| from k=1 to ∞ does not then 𝚺a(k) from k=1 to ∞ is conditionally convergent.

Question 40

The comparison test

Answer 40

Let 𝚺b(k) from k=1 to ∞, be a convergent series of non-negative numbers and suppose that for some constant M>0 we have |a(k)|≤Mb(k) for all k. Then the series 𝚺a(k) from k=1 to ∞ is absolutely convergent.

Question 41

The alternating series test

Answer 41

Let a(k) be a non-increasing sequence of positive numbers such that a(k)→0 as k→∞. Then the series 𝚺a(k)(-1)^(k+1) from k=1 to ∞ converges.

Question 42

How would you negate: P⇒Q ?

Answer 42

not-Q & P

Question 43

How would you negate: (A∩B) ?

Answer 43

not-A OR not-B

Question 44

How would you negate: A⇔B ?

Answer 44

(not-A & B) OR (A & not-B)

Question 45

How would you negate: (A∪B) ?

Answer 45

not-A OR not-B

Question 46

|A|<1 ⇒A^n→? as n→∞

Answer 46

0

Question 47

True or False:
If a(k)→0 as k→∞, then the series 𝚺a(k) from k=1 to ∞ must converge.

Answer 47

FALSE

Although it is true that if 𝚺a(k) from k=1 to ∞ converges, then one must have If a(k)→0 as k→∞.