1 / 26

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.

cai
Download Presentation

Foundations of Real Analysis

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Foundations of Real Analysis Sets and Sentences Open Sentences

  2. 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

  3. 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

  4. 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

  5. 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

  6. 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

  7. Example #2 Let E = {even integers}, O = {odd integers}, Z = {all integers}. Find each union, intersection, or complement. 6. O′

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

  9. 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

  10. 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.

  11. 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.

  12. 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.

  13. Example #8 List all subsets of each set. 30. {1, 3}

  14. Example #8 The 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}

  15. 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

  16. 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

  17. Conjunctions • If 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.

  18. Disjunctions • If p and q each represent sentences, then the disjunction of p and q is the sentence p or q, also written as • The conjunction porq 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.

  19. 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.

  20. Example #9 State whether the statement is true or false. 2. 3 is negative or 3 is positive

  21. Example #10 Find and graph the solution set over . a. p b. q c. 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

  22. Example #11 Find and graph the solution set over . a. p b. q c. 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

  23. Example #12 Write the negation of each sentence. 18. For every real number x, x > 0 or x < 0.

  24. 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

  25. Example #14 State whether each sentence over is an identity, a contradiction, or a sentence that is sometimes true and sometimes false. 32.

  26. Homework • Review notes • Complete Worksheet #1

More Related