## 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, Exercises 3,5,9,11)

• 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, Exercises 3,7,11,17)

• 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)"
(Exercises 7,9,23,35,43)

• 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). (Exercises 5,11,17,21).

Material from Midterm II

• 2.3 Understand the "Fundamental Principle of Counting."
Example 1, 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 3, Example 4. Problems 19,20.

• 2.4 Understand the difference between "permutation" and "combination."
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.
Example 2.
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.

New Material on Final Examination
• 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). Understand 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 3-state case.

• 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.

December 6 1999