1 / 40

Homework

Homework. Review notes Complete Worksheet #1. 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. Homework. Let E = {even integers}, O = {odd integers}, Z = {all integers}

alexa
Télécharger la présentation

Homework

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. 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 21 = 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 < 6 → p: x > 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 9 p: 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------→ ←------● 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 < -4 or 3x > 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. Foundations of Real Analysis Conditional Sentences Addition and Multiplication Properties of Real Numbers

  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 x2 < 0, then x4 ≥ 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: • a = b if and only if a + c = b + c Cancellation Law of Addition • a = b if and only if ac = bc (c ≠ 0) Cancellation Law of Multiplication • If a = b, – a = – b • – ( – a) = a • a∙0 = 0 • – 0 = 0 • – a = – 1(a)

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

  35. Theorem Two For all real numbers a and b: ab = 0 if andonly 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 a2 = b2 Proof: a = b aa = ab ab = bb aa = bb a2 = b2

  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

More Related