Download Foundations of Real Analysis

Foundations of Real Analysis
Sets and Sentences
Open Sentences
Sets
 Set – collection of objects for which there is a definite
criterion for membership and non-membership, usually
denoted by upper-case letter
 Member – object in a set, usually denoted by a lower
case letter
Symbology
 a  A, a is a member of A
 b  B , b is not a member of B
 N, the set of natural numbers
 Z, the set of integers
 Q, the set of rational numbers
  , the set of real numbers
Symbology
 G  H , G is a subset of H
 F  H , F is a proper subset of H, that is H has one or




more members not contained in F
If J = K, then J  K and K  J
 , the null or empty set, it has no members
A B , the union of A and B, all members of A plus all
members of B
A B , A intersect B, all members of A that are also
members of B
More Set Terminology
 Universal set – set with a large number of members,
such as the set of all real numbers or of all points on a
plane
 Complement of a set – those members of the universal
set not in the specified set, e.g., if A is a set and U is
the universal set, A′ is the complement of A, that is all
members of U not in A
Example #1
Let A = {a, b, c, d}, B = {a, b, c, d, e}, C = {a, d},
D = {b, c}
Describe any subset relationships.
2. C; D
Example #2
Let E = {even integers}, O = {odd integers}, Z = {all
integers}. Find each union, intersection, or complement.
6.
O′
Example #3
State whether each statement is true or false.
10. 21  all integral multiples of 7
Example #4
If A = {1, 2, 3, 4}, B = {1, 4, 6, 8}, C = {2, 4, 5, 8, 10}
and U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}, find:
14.
B∩C
Example #5
If A = {1, 2, 3, 4}, B = {1, 4, 6, 8}, C = {2, 4, 5, 8, 10}
and U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}, find:
18.  A C 
Example #6
If A = {1, 2, 3, 4}, B = {1, 4, 6, 8}, C = {2, 4, 5, 8, 10}
and U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}, find:
22. C  B C 
Example #7
If A = {1, 2, 3, 4}, B = {1, 4, 6, 8}, C = {2, 4, 5, 8, 10}
and U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}, find:
26.  A
Example #8
List all subsets of each set.
30.
{1, 3}
Example #8
The power set of a set A, denoted byP  A  , is the set of
all subsets of A. Tell how many members the power set
of each set has.
33.
{1, 3}
Sentences
 Sentences occur frequently in mathematics
 For instance: 3•4 = 12 is a true sentence, while 7 = 14 is a
false sentence.
 Let D be a set and let x represent any member of D.
 Any sentence involving x is an open sentence.
 x is the variable
 D is called the domain or replacement set of x
Identities
 An identity is an open sentence whose solution set is
the domain of its variables.
 For instance, x + 3 = 3 + x is an identity over the set of
real numbers
 A contradiction is a sentence whose solution set is
empty.
 For instance, x + 3 = 5 + x is a contradiction because no
real number satisfies x + 3 = 5 + x
Conjunctions
 If p and q each represent sentences, then the conjunction of
p and q is the sentence p and q, also written as p  q
 The conjunction p and q is true if both p and q are true and
false otherwise. It is sometimes displayed in a truth table.
q
true
pq
true
true false
false true
false false
false
false
false
p
true
 The solution set of the conjunction of two open sentences
is the intersection of the solution sets of the open sentences.
Disjunctions
 If p and q each represent sentences, then the disjunction of
p and q is the sentence p or q, also written as p  q
 The conjunction p or q is true if either p or q is true and
false otherwise. it is sometimes displayed in a truth table.
q
true
pq
true
true false
false true
false false
true
true
false
p
true
 The solution set of the conjunction of two open sentences
is the union of the solution sets of the open sentences.
Negations
 Consider the sentences: “1 = 0” and “1 ≠ 0.” The
second sentence is the negation of the first.
 If p is a sentence, then the sentence not p, also written
p′ is called the negation of p.
 Not p is true when p is false and false when p is true.
Example #9
State whether the statement is true or false.
2.
3 is negative or 3 is positive
Example #10
Find and graph the solution set over  .
a. p b. q c. p  q
6. p : x  0; q : 3x  15
-----|----|----|----|----|----|----|----|----|----|----|----|----|----|----|----7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7
-----|----|----|----|----|----|----|----|----|----|----|----|----|----|----|----7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7
-----|----|----|----|----|----|----|----|----|----|----|----|----|----|----|----7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7
Example #11
Find and graph the solution set over .
a. p b. q c. p  q
12. p : x  0; q : 3x  12
-----|----|----|----|----|----|----|----|----|----|----|----|----|----|----|----7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7
-----|----|----|----|----|----|----|----|----|----|----|----|----|----|----|----7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7
-----|----|----|----|----|----|----|----|----|----|----|----|----|----|----|----7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7
Example #12
Write the negation of each sentence.
18.
For every real number x, x > 0 or x < 0.
Example #13
26.
Find and graph on a number line the solution set
over  of the conjunction 2 x  4 and 3x  6.
-----|----|----|----|----|----|----|----|----|----|----|----|----|----|----|----7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7
Example #14
State whether each sentence over  is an identity, a
contradiction, or a sentence that is sometimes true and
sometimes false.
32. 2 x  1  5 and 3 x  2  11
Homework
 Review notes
 Complete Worksheet #1