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Delta
Epsilon
Pi
Lambda
Gamma
100
True or False? The intersection of A and B is the set A ∩ B = {x : x in A or x in B}.
False: This is the definition for the UNION of A and B.
100
Suppose f:D→R is continuous at every point on its domain. Furthermore, let c∈D be an accumulation point of D. Then what is the limit of f(x) as x→c?
f(c)
100
A sequence that is either non-increasing or non-decreasing is a _______________ sequence.
Monotone.
100
What kind of statement is false in all cases?
A contradiction.
100
True or False? A function is a bijection iff it is one-to-one.
False: A bijection is one-to-one (injective) AND onto (surjective).
200
When is l’Hôpital’s rule used, and what is the basic procedure?
L’Hôpital’s rule uses derivatives to evaluate limits involving indeterminate forms. If we are taking the limit as x→∞ of f(x)/g(x), we simply take the derivative of the numerator f’(x) and divide it by the derivative of the denominator g’(x). Thus, we would have f’(x)/g’(x). If this quotient results in another indeterminate form, the process is simply repeated over and over until a determined limit is reached.
200
Which proof technique would be most appropriate to prove that the interval (0,1) is uncountable?
Proof by contradiction.
200
A sequence is a function from the _____ to the Reals.
Natural Numbers.
200
True or False: If a non-empty set of real numbers is bounded above, then it has a supremum.
True (Completeness Axiom).
200
The set of all interior points of S is denoted by ∂S.
False = int S.
300
If sn is an unbounded non-decreasing sequence, then lim sn=_____
+∞
300
True or False: If a sequence converges to a real number s, then every subsequence of the sequence also converges to s.
True!
300
True or False: Let f and g both be continuous functions from D to the reals. Suppose they are both continuous at c. Then, f/g is always continuous at c as well.
FALSE: f/g continuous at c ONLY IF g(c) is non-zero.
300
A point x is a boundary point of S if for every neighborhood N of x, N∩S=∅ and N∩(R\S)=∅.
False: N∩S and N∩(R\S) do NOT equal the empty set.
300
Let A = {1, 3, 4, 5, 7} and B = {3, 4, 5, 6} find A \B.
A\B = {1,7}
400
What is a tautology?
A tautology is a statement that is true in all cases.
400
Give an example of a periodic function.
Many possibilities. Ex: Trigonometric functions.
400
Given the sequence sn= (3n+1)/(n+2), find lim(sn).
The limit of the sequence is 3.
400
What is the limit of sin(1/x)as x→∞?
The limit does not exist.
400
What is the most direct way to prove an existence statement (i.e. There exists an x such that ...)
The most direct way is to construct or find an x that satisfies the property stated.
500
What is the relationship between the Cartesian product A x B and a relation from A to B?
A relation from A to B is a subset of the Cartesian product A x B.
500
State the Intermediate Value Theorem in your own words (i.e. you don’t have to state the formal theorem but must explain the basic concept).
The intermediate value theorem basically states that if a function is continuous on the closed interval [a,b], then if k falls somewhere between f(a) and f(b), then there exists some point c in the interval [a,b] such that f(c) = k.
500
Name the following axiom: Given a collection of non-empty pairwise disjoint sets, there exists a set containing exactly one element taken from each set.
This is called the Axiom of Choice.
500
When proving a field is ordered, we must consider 4 order axioms. List and briefly explain these 4 properties.
1. Trichotomy. Exactly one of the following is true: xx, x=y 2. Transitivity. If x
500
Let D be the open interval (0,5). Give a continuous function that is not bounded on D.
MANY options! An example is: f(x)=1/(x(x-5)) because the denominator will be 0 when x=0 and when x=5.