First Last
Math 270A
12/10/13
Assignment #22: Section 11.3: Problems 2, 9, 11, 13, 19
2. One can show that the Associative and Distributive Laws hold for lcm and gcd directly. The
Commutative Law clearly holds. To see that the Identity Laws hold, note tha
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
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
Fall 2013
Math 270A
11.4 Boolean Functions and Synthesis of Circuits
1
11.4 Boolean Functions and Synthesis of Circuits
A circuit is constructed to carry out a specied task. If we want to construct a combinatorial
circuit, the problem can be given in term
Fall 2013
Math 270A
6.8 The Pigeonhole Principle
1
6.8 The Pigeonhole Principle
The Pigeonhole Principle (also known as the Dirichlet Drawer Principle or the Shoe Box
Principle is sometimes useful in answering the question: Is there an item having a given
Fall 2013
Math 270A
11.5 Applications
1
11.5 Applications
In the preceding section we showed how to design a combinatorial circuit using AND, OR, and
n
NOT gates that would compute an arbitrary function from Zn into Z2 , where Z2 = cfw_0, 1. In
this secti
Fall 2013
Math 270A
1.1 Sets
1
1.1 SETS
The concept of set is basic to all of mathematics and mathematical applications. A set is simply
a collection of objects. The objects are sometimes referred to as elements or members. If a set is
nite and not too la
Fall 2013
Math 270A
11. 1 Combinatorial Circuits
1
11. 1 Combinatorial Circuits
In a digital computer, there are only two possibilities, written 0 and 1, for the smallest, indivisible object. All programs and data are ultimately reducible to combinations
Fall 2013
Math 270A
1.5 Quantiers
1
1.5 Quantiers
Denition 1.5.1 Let P (x) be a statement involving the variable x and let D be a set. We
call P a propositional function or predicate (with respect to D) if for each x D, P (x) is a
proposition. We call D t
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
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
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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
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
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
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 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
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
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
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,
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
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
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
Math 270A
11/14/13
Assignment #16: Section 6.2: Problems 14ALL, 1024 even, 3448 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,
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
Math 270A
Name_
Quiz 7
1. Determine whether the following relation, defined on the set of positive integers, is:
a) reflexive
b) symmetric
c) antisymmetric
d) transitive
e) a partial order
f) an equivalence relation
Prove your answers.
(, ) if 3 divides
Math 270A
Name_
Quiz 3
(5 points)
1. Write the following argument symbolically and determine whether it is valid
or invalid.
If I study hard or I get rich, then I get As.
I get As.

If I dont study hard, then I get rich.
(5 points)
2. Let () denote the
Math 270A
Name_
Quiz 4
(5 points)
1. Determine the truth value of the given statement.
The domain of discourse is . Justify your answer.
( < ) ( 2 < 2 )
(5 points)
2. If the statement is true, prove it; otherwise, give a counterexample.
( ) ( ) = for all
Math 270A
Name_
Quiz 5
(5 points)
1. Given () = 6 9 with Domain = and Codomain = .
Is () onetoone? Is () onto? Prove your answers.
(5 points)
2. Given () = 3 2 3 + 1 with Domain = and Codomain = .
Is () onetoone? Is () onto? Prove your answers.