Fall 2013
Math 270A
11.3 Boolean Algebras
1
11.3 Boolean Algebras
In this section we consider general systems that have properties like those given in Theorem
11.2.1. We will see that apparently systems obey these same laws. We call such systems Boolean
a
Fall 2013
Math 270A
Exam 2
Name:
Number:
(30 points) 1. Classify each of the following statements as always, sometimes, or never true.
(1)
The codomain of a function f is the same set as the range of f .
(2)
A function is a set.
(3)
The oor of x, is the l
First Last
Math 270A
11/14/13
Assignment #16: Section 6.2: Problems 1-4ALL, 10-24 even, 34-48 even
1. How many permutations are there of a, b, c, d?
= 4!
= 24
2. List the permutations of a, b, c, d.
= abcd, abdc, acbd, acdb, adbc, adcb, bacd, badc, bcad,
First Last
10/01/13
Math 270A
Assignment #7: Section 2.1: Problems 9, 13, 16, 24, 39, 54, 55
9. If m and n are even integers, then m = 2a and n = 2b.
Then mn = (2a)(2b) = 4ab = 4d.
4d = 2(2d), which must be an even number.
Therefore, for all m and n, mn i
First Last
10/10/13
Math 270A
Assignment #9: Section 2.4: Problems 3, 5, 6, 23
3. Basis step (n=1)
For n = 1: 1(1!) = (1+1)!-1 1 = 1, which is true.
Inductive step
Assume the statement is true for n.
Now, n+1(n+1)!) = (n+2)!-1
Since (n+1)! = (n+1)(n!), we
First Last
MATH 270A
10/17/13
Assignment #10: Section 3.1: Problems 11, 15, 18, 36, 39, 43, 46
11. WNTS that for all n on the set of all integers and for all n2 on the set of all integers, that if n1
=/= n2, then f(n1) =/= f(n2).
Suppose n1, n2 on the set
First Last
Math 270A
11/21/13
Assignment #19: Section 6.8: Problems 2, 6, 8, 10, 11
2. 6 students, 2 received same grade
Since there are only 5 different grades, there are too many students to receive a unique grade
S1 = A, S2 = B, S3 = C, S4 = D, S5 = F,
Fall 2013
Math 270A
1.2 Propositions
1
1.2 Propositions
A sentence that is either true or false, but not both, is called a proposition. A proposition
is typically expressed as a declarative sentence (as opposed to a question, command, etc.). Propositions
Fall 2013
Math 270A
6.1 Basic Principles
1
6.1 Basic Principles
The menu for Kays Quick Lunch is shown in gure below. As you can see, it features two
appetizers, three main course, and four beverages. How many dierent dinners consist of one
main course sa
Fall 2013
Math 270A
Exam 3
Name:
Number:
(30 points) 1. Classify each of the following statements as always, sometimes, or never true.
(1)
If we are counting objects that are constructed in successive steps, we use the Multiplication Principle.
(2)
If we
First Last
Math 270A
10/08/13
Assignment #8: Section 2.2: Problems 18, 28, 30, 38, 42, 44
18. Assume all nine boxes contain less than 11 balls, which is less than 12.
If each box contains 11 balls, then the total amount of balls would be equal to 99.
Howe
Fall 2013
Math 270A
Exam 1
Name:
Number:
(30 points) 1. Classify each of the following statements as always, sometimes, or never true. Assume that
A and B are arbitrary set.
(1)
If |A| = |B|, then A = B.
(2)
If A B, then |A| |B|.
(3)
If A B, then |A| = |B
Fall 2013
Math 270A
1.3 Conditional Propositions and Logical Equivalence
1
1.3 Conditional Propositions and Logical Equivalence
The dean has announced that
If the Mathematics Department gets an additional $60,000,
then it will hire one new faculty member.
Fall 2013
Math 270A
6.7 Binomial Coecients and Combinatorial Identities
6.7 Binomial Coecients and Combinatorial Identities
Binomial Theorem
If a and b are real numbers and n is a positive integer, then
n
n
C(n, k)ank bk .
(a + b) =
k=0
The numbers C(n, r
Fall 2013
Math 270A
11.2 Properties of Combinatorial Circuits
1
11.2 Properties of Combinatorial Circuits
In the preceding section we dened two binary operators and on Z2 = cfw_0, 1 and a unary
operator on Z2 . (Through out the remaining of this chapter w
Fall 2013
Math 270A
1.4 Arguments and Rules of Inference
1
1.4 Arguments and Rules of Inference
Consider the following sequence of propositions.
The bug is either in module 17 or in module 81.
The bug is a numerical error.
Module 81 has no numerical error
MATH 270A Mathematical Structures I
MW 4:00 pm 5:15 pm
Spring 2015
Michael Campbell
Email:
SECTION 18993
CSUF room MH 487
ver 2015.01.29
Office:
. Office Hours:
MH 612B
W 2:553:55pm
Th 2:00-3:00pm
COURSE INFORMATION
Discrete Mathematics, 7 ed. by Richard
Fall 2013
Math 270A
6.5 Introduction to Discrete Probability
1
6.5 Introduction to Discrete Probability
Probability was developed in the seventeenth century to analyze games and, in this earliest
form, directly involved counting. For example, suppose that
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Fall 2013
Math 270A
6.2 Permutations and Combinations
1
6.2 Permutations and Combinations
Denition 6.2.1 A permutation of n distinct elements x1 , . . . , xn is an ordering of the n
elements x1 , . . . , xn .
Example 6.2.2 Find the permutations of the ele
Fall 2013
Math 270A
6.3 Generalized Permutations and Combinations
1
6.3 Generalized Permutations and Combinations
In Section 6.2, we dealt with ordering and selections without allowing repetitions. In this section we
consider orderings of sequences contai