Foundations of Real Analysis

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