2.3 Extended Set Operations and Indexed Families
of Sets.
A. Families of Sets
1. Definition 1. A set of sets is called a
family of sets. If is such a family then
2. Example.
is a family of sets. Note that
3. Example. Given a set
family of sets.
is a
Recal
MATH 290 WRITING ASSIGNMENT 8 SOLUTIONS
1. Problem 5 p. 171.
Solution:
In order to show that the relation R is a partial order, we must show that it is
reexive, antisymmetric and transitive. Let x N. Then x = 20 x so that xRx
since 0 Z and 0 0. Hence R is
MATH 290 WRITING ASSIGNMENT 7 SOLUTIONS
1. Problem 19(a), (b), and (d), p. 173.
Solution:
Let A be a set with partial order R and for each a A, dene the set Sa = cfw_x
A: xRa.
(a). Let a, b A and assume that aRb. We must show that Sa Sb . Let x Sa .
This
MATH 290 WRITING ASSIGNMENT 9 SOLUTIONS
1. Prove that if xn L and yn M , then xn yn LM .
Solution:
There is more than one way to prove this. The rst proof will use the fact that
a convergent sequence is bounded. Specically we state: Lemma 1. If yn M
then
MATH 290 WRITING ASSIGNMENT 5 SOLUTIONS
1. Use a form of induction to prove that for all natural numbers n 2, either n is prime or
n has a prime factor p with p n.
Solution:
The proof will use the Principle of Complete Induction.
Assume that n = 2. Since
MATH 290 EXAM 1 SOLUTIONS
1. (5 pts. each)
(a) Is [Q (P = Q)] = P a tautology, contradiction, or neither? Justify your answer
with a truth table or portion of a truth table.
(b) Give a useful denial in propositional form of P Q = R.
Solution:
(a). The exp
MATH 290 WRITING ASSIGNMENT 3 SOLUTIONS
1. Prove that for any sets A and B, (A B) (B A) = (A B) (A B).
Solution:
Let A and B be sets. Assume rst that x (A B) (B A). We will show
that x (A B) (A B). Since x (A B) (B A), x A B or
x B A. This means that ther
MATH 290 WRITING ASSIGNMENT 2 SOLUTIONS
1
1
1. Prove that for every real number x > 0, x + 2 and that x + = 2 if and only if
x
x
x = 1.
Solution:
1
Let x > 0. We must show that x + 2. Since the square of any real number
x
is nonnegative, (x 1)2 0. Multipl
MATH 290 EXAM 2 SOLUTIONS
1. (5 pts.) Let a1 = 3, a2 = 9, and for n 3, an = 5an1 6an2 . Use the Principle of
Complete Induction to prove that for all n N, an = 3n .
Solution:
We will prove this using the Principle of Complete Induction. If n = 1 or n = 2
MATH 290 WRITING ASSIGNMENT 4 SOLUTIONS
1. Prove that for any sets A and B, (A B) (B A) = (A B) (A B).
Solution:
Let A and B be sets. Assume that (x, y) (A B) (B A). We must show
that (x, y) (A B) (A B). Since (x, y) (A B) (B A), (x, y) (A B)
and (x, y) (
MATH 290 WRITING ASSIGNMENT 6 SOLUTIONS
1. Let a and b be natural numbers, and let d be the smallest natural number such that there
exist integers x and y such that ax + by = d. Prove that d = GCD(a, b). (Hint: Letting
c = GCD(a, b), rst prove that c d by
1.2 Conditionals and Biconditionals
A. Conditional:
1. Read
implies
If
then
whenever
only if
is sufficient for
is necessary for
T
T
F
F
T
F
T
F
Why is
2.
T
F
T
T
true when
is equivalent to
T
T
T
F
T
F
F
F
T
F
T
T
is false?
3. Definition 1. is t
2.1 Basic Concepts of Set Theory.
A. Belonging
1. A set is thought of as a collection of
elements.
2. A fundamental property of a set is
determining whether an element does or
does not belong to . If it does, we write
. So given , either
or
.
3. Typically
2.4/2.5. Mathematical Induction.
A.
Idea behind induction
1. Suppose you have an open sentence
defined for all
, and you want to prove
.
2. Induction says you can do this as follows:
a. Prove
. This is called the base case.
b. Prove that
.
This is called
3.1. Cartesian Products and Relations
A. Cartesian Products
1. Definition 1. Let
and
Cartesian product of
is the set
2.
be sets. The
and , denoted
,
is an ordered pair, meaning that it is a
two-element set in which the order matters.
Given two ordered pai
3.2. Equivalence Relations.
A.
Definition and Examples
1. Definition 1. Let
is reflexive if
is symmetric if
is transitive if
be a relation on a set .
is an equivalence relation if it is reflexive,
symmetric and transitive.
2. Example. The identity relatio
3.4. Ordering Relations.
A.
Partial Orderings
1. Idea: Partial orders stand in relation to
in the same way that equivalence relations
stand in relation to .
2. Properties of .
a. Reflexive?
b. Symmetric?
c. Transitive?
3. Example. divides on .
4. Example
3.3. Partitions.
A.
Definition and Examples
1. Definition 1. Let be a nonempty set. A
partition of is a collection of subsets of
satisfying
a. if
b. if
c.
, then
and
, and
.
then either
or
2. Partitions and Equivalence Relations.
a. Example. Consider the
1.4-1.7
Proofs and Proof Methods
A. Basic assumptions about
1.
2. Each is closed under addition (+) and
multiplication ( ) and the usual
commutative, associative, and distributive
laws hold.
3. For
,
a. there is a unique number such that for
all ,
(additi
2.2 Set Operations.
A. Intersections, Unions, Differences.
1. Definition 1. Let
and
be sets.
i.
If
then
and
If
then
and
If
then
and
.
ii.
.
iii.
.
B. Proving Theorems involving sets.
Theorem 2.6. Let , , and
i.
ii.
iii.
iv. If
Proof:
then
be sets.
C. Russ
1.1 Propositions and Connectives
1. Deductive Reasoning
a. Drawing correct conclusions from
assumptions.
b. Conclusions are logically necessary.
c. Conclusions must be correct.
2. Inductive Reasoning
a. Postulating general principles based on
observations
1.3 Quantifiers
A. Open Sentences
1. Recall that
is not a proposition.
On the other hand, if we write
Then for each value of ,
is a proposition.
is called an open sentence.
2. Definition 2. The truth set of an open
sentence
is the set of all such that
i
MATH 290 WRITING ASSIGNMENT 1 SOLUTIONS
1. Prove that for all real numbers a and b, |a| b if and only if b a b.
Solution:
Let a and b be real numbers.
(=). Assume that |a| b. We will consider two cases. Assume rst that a 0. In
this case, |a| = a. Since |a