1. Homework Review notes Complete Worksheet #1

2. Homework Let A = {a,b,c,d}, B = {a,b,c,d,e}, C = {a,d}, D = {b, c} Describe any subset relationships. 1. A; D

3. Homework Let E = {even integers}, O = {odd integers}, Z = {all integers} Find each union, intersection, or complement. 5.

4. Homework State whether each statement is true or false. 9.- False

5. Homework 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: 13.

6. Homework 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: 17.A’

7. Homework 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: 21.

8. Homework 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: 25.

9. Homework List all subsets of each set. 29.{4}

10. Homework The power set of a set A, denoted by P (A) is the set of all subsets of A. Tell how many members the power set of each set has. 33.{4} The power set of A has 2 1 = 2 members

11. Homework State whether each statement is true or false. 1.4 is an even number and 5 is an odd number – True

12. Homework Find and graph each solution set over R; i.e., p, q, and p Λ q 5.p: x > 0; q: 2x 0; q: x < 3 ο-------------ο ο---→ ←---ο -----|----|----|----|----|----|----|----|----|----|----|----|----|----|----|---- -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

13. Homework Find and graph each solution set over R; i.e., p, q, and p Λ q 9p: 4t – 5 ≥ 3; q: 3t + 5 ≤ 26 → p: 4t ≥ 8; q: 3t ≤ 21 → p: t ≥ 2; q: t ≤ 7 ●---------------------● ●------→ ←------● -----|----|----|----|----|----|----|----|----|----|----|----|----|----|----|---- -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

14. Homework Find and graph each solution set over R; i.e., p, q, and p ν q 13. p: 3w – 1 > 5; q: 4w +3 ≤ -1 → p: 3w > 6; q: 4w ≤ - 4 → p: w > 2; q: w ≤ -1 ←------● ο------→ -----|----|----|----|----|----|----|----|----|----|----|----|----|----|----|---- -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

15. Homework Write the negation of each sentence. 17.There is a positive square root of 2. There is not a positive square root of 2.

16. Homework Write the negation of each sentence. 21.

17. Homework 25. Find and graph on a number line the solution set over R of the negation of the conjunction 2x 6 → 2x ≥ -4 and 3x ≤ 6 → x ≥ -2 and x ≤ 2 ●------------● -----|----|----|----|----|----|----|----|----|----|----|----|----|----|----|---- -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

18. Homework State whether each sentence over is an identity, a contradiction, or a sentence that is sometimes true and sometimes false. 29.

19. Homework State whether each sentence over is an identity, a contradiction, or a sentence that is sometimes true and sometimes false. 33.

20. Conditional Sentences Addition and Multiplication Properties of Real Numbers Foundations of Real Analysis

21. Conditional Sentence Conditional sentence – sentence in which there is a dependency of one sentence on another; if p and q are sentences, a conditional sentence relating them is “if p, then q” (p → q) Conditional sentences, by definition, are always true except when p is true and q is false Converse – the opposite dependency of a conditional sentence, the converse of p → q is q → p (“if q, then p”) Biconditional sentences are true only when both p and q are true or both p and q are false Contrapositive – statement q’ → p’ is the contrapositive of p → q

22. Example #1 State whether the conditional sentence is true or false 2.If 12 is a multiple of 6, then 24 is a multiple of 6

23. Example #2 Give the converse of the conditional sentence and state if it is sometimes, always, or never true. 6.If 2 is a factor of an integer, then 2 is a factor of the square of that integer.

24. Example #3 Give the converse of the conditional sentence and state if it is sometimes, always, or never true. 10. If x 2 < 0, then x 4 ≥ 0

25. Example #4 State the contrapositive for each conditional sentence. 14.If ab = ac and a ≠ 0, then b = c.

26. Formal Mathematical Systems A formal mathematical system consists of: Undefined objects Postulates or axioms Definitions Theorems

27. Axioms of Equality Axioms of Equality (for all real numbers a, b, and c) : Reflexive Property: a = a Symmetric Property: If a = b, then b = a Transitive Property: If a = b and b = c, then a = c

28. Substitution Axiom Substitution Axiom: If a = b, then in any true sentence involving a, we may substitute b for a, and obtain another true sentence

29. Axioms of Addition Closure For all real numbers a and b, a + b is a unique real number Associative For all real numbers a, b, and c Additive Identity There exists a unique real number 0 (zero) such that for every real number a. Additive Inverses For each real number a, there exists a real number – a (the opposite of a) such that Commutative For all real numbers a and b,

30. Axioms of Multiplication Closure For all real numbers a and b, ab is a unique real number Associative For all real numbers a, b, and c Multiplicative Identity There exists a unique real number 1 (one) such that for every real number a. Multiplicative Inverses For each real number a, there exists a real number (the reciprocal of a) such that Commutative For all real numbers a and b,

31. Distributive Axiom of Multiplication over Addition For all real numbers a, b, and c,

32. Definitions Subtraction : Division: provided b ≠ 0

33. Theorem One For all real numbers a, b, and c: 1. a = b if and only if a + c = b + c Cancellation Law of Addition 2. a = b if and only if ac = bc (c ≠ 0) Cancellation Law of Multiplication 3. If a = b, – a = – b 4. – ( – a) = a 5. a∙0 = 0 6. – 0 = 0 7. – a = – 1(a)

34. Theorem One Continued For all real numbers a, b, and c: 8. – ab = a (– b) = – a (b) 9. – (a + b) = – a + ( – b) 10. If a ≠ 0,

35. Theorem Two For all real numbers a and b: ab = 0 if and only if a = 0 and/or b = 0

36. Example #5 Name the axiom, theorem, or definition that justifies each step. 2.If a = b, then a 2 = b 2 Proof: a = b aa = ab ab = bb aa = bb a 2 = b 2

37. Example #6 Solve over R. 6.

38. Example #7 Solve over R. 10.

39. Example #7 State whether each set is closed under (a) addition and (b) multiplication. If not, give an example. 14.{1}

40. Homework Review notes Complete Worksheet #2