MAT118 Final Review (corrected, Dec. 6 afternoon)
Material from Midterm I
1.1 Be able to analyze a syllogism with a Venn diagram. (Examples 2,3,4,
- 1.2 Understand the definition of "statement" and the use of the logical
connectives "and" "or" "not" and "implies". Be able to
translate a statement into a symbolic representation. (Examples 3,4,5,
- 1.3 Use truth tables to figure out under what circumstances a certain
statement is true (Examples 3,4). Understand what it means for two
statements to be equivalent, and be able to use truth tables to establish
equivalence of statements (Example 5). Understand de Morgan's laws:
- "not (A and B)" is equivalent to "(not A) or (not B)"
- "not (A or B)" is equivalent to "(not A) and (not B)"
- 1.4 Understand the relation between a conditional and its
converse, inverse and contrapositive. Understand how to show
(using truth table, for example) that a conditional and its contrapositive
are equivalent, and that its converse and inverse are equivalent, but
that a conditional and its inverse are not equivalent (Example 3).
Understand that in "if A then B" A is "the premise" and B is " the
conclusion" (Example 4).
Understand "only if": "A only if B" is equivalent to "if A then B"
and understand that "A if and only if B" means "A implies B and
B implies A" (Example 5). (Exercises 7,13,21,27,35)
- 1.5 Be able to use a truth table to analyze an argument (Examples 1,3,4).
Material from Midterm II
New Material on Final Examination
- 2.3 Understand the "Fundamental Principle of Counting."
Problems 4 and 5.
Understand the factorial 5! = 5 4 3 2 1 notation and how to
use it in counting how many ways n objects can be ordered.
Example 4. Problems 19,20.
- 2.4 Understand the difference between "permutation" and
Example 2, Problem 18.
Example 5, Example 8, Problem 35.
- 3.2 Understand the "basic probability terms" and know the definition
of "Probability of an Event" (p.122).
Example 3, Problems 6,7,8,9.
Understand how probabilities occur in genetics.
Example 4, Problems 61,62,63.
- 3.3 Understand the "probability rules" (p.135 and p.139) and be able to tell
if two events are mutually exclusive.
Especially Rule 4:
p(EUF) = p(E) + p(F) - p(E^F). Know how to interpret this rule in terms
of a Venn Diagram.
Example 4, Problems 60,61.
- 3.4 Understand how "combinations" enter into calculating probabilities.
Example 3 and Example 4 <--understand these! Problems 15,16,19,20.
- 3.5 Know the definition of "Expected Value" and how to compute it.
Example 1 is a good one. Problems 14,15.
Harder problems like 20,21,22
are worth knowing. Problem 23.
- 3.6 Know the definition of "Conditional Probability" and how to compute
it as in Example 1.
Understand the rule p(A^B) = p(A)p(B|A) and how to
use it as in Example 2.
Problems 3-6, Problems 33-36.
- 3.7 Understand ``independence'' (this is important!):
Events A and B are independent if p(A^B)=p(A)p(B). Equivalently,
p(A|B) = p(A) or p(B|A) = p(B); all three statements mean the same
thing. Example 1, Example 2 are elementary, as are Problems
1-10; Problem 11 uses probability. This section also has important
applications of probability to problems in genetics (Example 6,
Example 7, Problems 24 and 25, also Problem 33) and to the
problem of "False positives" (and
negatives) when tests are applied. (Example 5, Problems 18,19,20).
- 7.0 Know when two matrices can be multiplied and how to
compute their product. This is worked out in detail for the 2 x 2
case in Example 1. Understand that matrix multiplication is
not commutative in general: Example 3. Be able to multiply
2 x 2 and 3 x 3 matrices by hand. Be able to calculate the
powers A2, A3 of a 2 x 2 matrix A.
You are not responsible
for the "Graphing Calculator" part of the section.
- 7.1 Understand the definition of a Markov chain (First
paragraph on p.464 and text in Example 1). Understand how
all the information about the chain is encoded in the
transition matrix T (Example 1, Example 2).
(Remember that the row gives the state you are coming from,
and the column gives the state you are going to).
what is meant by a probability matrix and that
the probability matrix P2
for the first
following state is calaulated from the
probability matrix P1 for the current state
by the matrix multiplication P2 = P1 T.
Example 5. Also understand Examples 6 and 7. Problems 7 and 13
give a typical 2-state case; Problems 11 and 15 give a typical
- 7.2 Be able to solve a system of 2 linear equations in
two unknowns, and a system of 3 linear equations in 3 unknowns.
Using the "Elimination Method" (Example 2) is fine and also
works fine in the 3 x 3 case. You are not responsible for the
material on the "Gauss-Jordan Method" or for the material
on "Technology." In the examples we will consider, these
calculations can be done by hand or with a simple +,-,x
calculator. Be able to do problems like 19-24 and 50-59
by "elimination" or by any other method.
- 7.3 Understand that the square T2 of
a transition matrix T contains the probabilities of
2-step moves (see the end of 7.1-Example 8 on p.474).
Understand that the Equilibrium Matrix L is
the probability matrix that satisfies L T = L, and
be able to calculate L given T (Example 1 for 2 x 2 case,
Example 2 for 3 x 3 case). Exercises 1-4 for the calculation.
Understand long-range prediction using Markov chains.
Exercises 5 and 6 for the 2 x 2 case, 8 and 9 for the
3 x 3 case.