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1. Homework Review notes Complete Worksheet #2

2. Homework State whether the conditional sentence is true or false 1. If 1 = 0, then 1 = – 1 True F F T

3. Homework Give the converse of the conditional sentence and state if it is sometimes, always, or never true. 5. If two triangles are congruent, then their corresponding angles are congruent. If the corresponding angles of two triangles are congruent, then the triangles are congruent. Sometimes true.

4. Homework Give the converse of the conditional sentence and state if it is sometimes, always, or never true. 9. If ab < ac, then b < c If b < c, then ab < ac, sometimes true

5. Homework State the contrapositive for each conditional sentence. 13. If an angle of a parallelogram is a right angle, then the parallelogram is a rectangle. If a parallelogram is not a rectangle, then no one of its angles is a right angle.

6. Homework State the assumptions that would have to be made if the given statement is to be proven by contradiction. 17.

7. Homework State each conditional sentence in the if-then form. 21.

8. Homework State each conditional sentence in the if-then form. 25.

9. Homework Name the axiom, theorem, or definition that justifies each step. 1. Proof: Definition of Subtraction Associative Axiom Additive Inverses Additive Identity Transitive Property

10. Homework Solve over 5.

11. Homework Solve over 9.

12. Homework State whether each set is closed under (a) addition and (b) multiplication. If not, give an example. 13. N (a) Closed (b) Closed

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

14. Order The real numbers are ordered by the relation less than (<). a < b if the graph of a is to the left of the graph of b on the number line. Less than (<) is an undefined relation.

15. Axiom of Comparison For all real numbers a and b,

16. Transitive Axiom of Order For all real numbers a, b and c, if a < b and b < c, then a < c

17. Addition Axiom of Order For all real numbers a, b and c, if a < b, then a + c < b + c

18. Positive and Negative A real number, x, is positive if x > 0 and negative if x < 0

19. Multiplication Axiom of Order For all real numbers a, b and c, if 0 < c and a < b, then ac < bc.

20. Sign Definitions • a > b if and only if b < a • a ≥ b if and only if a > b or a = b • a ≤ b if and only if a < b or a = b

21. Theorem Three For all real numbers a, b, and c:

22. Theorem Four For all real numbers a, b, and c, if c < 0 and a < b, then ac > bc

23. Theorem Five For all real number a and b: • ab > 0 if and only if a and b are both positive or a and b are both negative. • ab < 0 if and only if a and b are opposite in sign.

24. Example Solve the inequality and graph each non-empty solution. 2. -----|----|----|----|----|----|----|----|----|----|----|----|----|----|----|----

25. Example Solve the inequality and graph each non-empty solution. 6. -----|----|----|----|----|----|----|----|----|----|----|----|----|----|----|----

26. Example Solve the inequality and graph each non-empty solution. 10. -----|----|----|----|----|----|----|----|----|----|----|----|----|----|----|----

27. Example Solve the inequality and graph each non-empty solution. 14. -----|----|----|----|----|----|----|----|----|----|----|----|----|----|----|----

28. Example 18. Solve and graph the intersection of the sets: -----|----|----|----|----|----|----|----|----|----|----|----|----|----|----|----

29. Absolute Value Definition – For all real numbers x: This may also be written as a piecewise function:

30. Theorem Six If a ≥ 0, |x| = a if and only if x = –a or x = a. If a > 0, |x| < a if and only if –a < x < a If a > 0, |x| > a if and only if x < –a or x > a

31. Theorem Seven For all real numbers a, |a|2 = a2

32. Theorem Eight For all real numbers a and b,

33. Example Find and graph each non-empty set over R. 2. -----|----|----|----|----|----|----|----|----|----|----|----|----|----|----|----

34. Example Find and graph each non-empty set over R. 6. -----|----|----|----|----|----|----|----|----|----|----|----|----|----|----|----

35. Example Find and graph each non-empty set over R. 10. -----|----|----|----|----|----|----|----|----|----|----|----|----|----|----|----

36. Example Find and graph each non-empty set over R. 18. -----|----|----|----|----|----|----|----|----|----|----|----|----|----|----|----

37. Example Find and graph each non-empty set over R. 22. -----|----|----|----|----|----|----|----|----|----|----|----|----|----|----|----

38. Example Solve. 26.

39. Natural Numbers The set of all positive integers {1, 2, 3,…}

40. Factor or Divisor a is a factor of b (symbolically, a|b) if there is an integer c such that ac = b

41. Prime Number Any number greater than 1 that has only one and itself as factors. Sometimes called a prime.

42. Composite Any integer greater than 1 that is not prime

43. Theorem Nine The Fundamental Theorem of Arithmetic – Every integer greater than 1 can be expressed as a product p1p2p3…pn in which p1, p2, p3, …, pn are primes. Furthermore, the factorizing is unique, except for the order in which primes are written.

44. Greatest Common Factor The greatest common factor or GCF of a and b is the largest positive integer that is a factor of both a and b

45. Relatively Prime Integers whose GCF is 1

46. Theorem Ten The Division Algorithm – Given integers s and t, t > 0, there exist unique integers q and r such that s = tq + r and 0 ≤ r < t

47. Euclidean Algorithm To find the GCF, divide the larger by the smaller, then the divisor by the remainder and so on until the remainder is zero; the last divisor is the GCF For example:

48. Rational numbers

49. Irrational Numbers All real numbers that are not rational, e.g., π and e, the base for natural logarithms

50. Theorem Eleven There is no rational number whose square is 2.