Foundations of Real Analysis

Sets and SentencesOpen SentencesFoundations of Real AnalysisSetsSet collection of objects for which there is a definite criterion for membership and non-membership, usually denoted by upper-case letterMember object in a set, usually denoted by a lower case letter

Symbology , a is a member of A , 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 is a subset of H , F is a proper subset of H, that is H has one or more members not contained in F If J = K, then and , the null or empty set, it has no members , the union of A and B, all members of A plus all members of B , A intersect B, all members of A that are also members of B

More Set TerminologyUniversal 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 #1Let 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 #2Let E = {even integers}, O = {odd integers}, Z = {all integers}. Find each union, intersection, or complement.6.O

Example #3State whether each statement is true or false.10.

Example #4If 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 #5If 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.

Example #6If 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.

Example #7If 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.

Example #8List all subsets of each set.30.{1, 3}Example #8The power set of a set A, denoted by , is the set of all subsets of A. Tell how many members the power set of each set has.33.{1, 3}

SentencesSentences occur frequently in mathematicsFor instance: 34 = 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 variableD is called the domain or replacement set of x

IdentitiesAn 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

ConjunctionsIf p and q each represent sentences, then the conjunction of p and q is the sentence p and q, also written as 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.

The solution set of the conjunction of two open sentences is the intersection of the solution sets of the open sentences.

DisjunctionsIf p and q each represent sentences, then the disjunction of p and q is the sentence p or q, also written as 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.

The solution set of the conjunction of two open sentences is the union of the solution sets of the open sentences.

NegationsConsider 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 positiveExample #10 Find and graph the solution set over . a. pb. qc. 6.

-----|----|----|----|----|----|----|----|----|----|----|----|----|----|----|---- -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. pb. qc. 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

-----|----|----|----|----|----|----|----|----|----|----|----|----|----|----|---- -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.

HomeworkReview notesComplete Worksheet #1