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- 1. 1Venn diagramA UC B The universal set U is usually represented by a rectangle. Inside this rectangle, subsets of the universal set are represented bygeometrical gures.

2. 2Venn diagrams help us identify some useful formulas in set operations. 3. 2Venn diagrams help us identify some useful formulas in set operations. To represent (A B) C : To represent (A C) (B C): 4. 2Venn diagrams help us identify some useful formulas in set operations. To represent (A B) C :ACB To represent (A C) (B C): 5. 2Venn diagrams help us identify some useful formulas in set operations. To represent (A B) C :ACB To represent (A C) (B C): 6. 2Venn diagrams help us identify some useful formulas in set operations. To represent (A B) C :ACB To represent (A C) (B C): 7. 2Venn diagrams help us identify some useful formulas in set operations. To represent (A B) C :ACB To represent (A C) (B C):ACB 8. 2Venn diagrams help us identify some useful formulas in set operations. To represent (A B) C :ACB To represent (A C) (B C):ACB 9. 2Venn diagrams help us identify some useful formulas in set operations. To represent (A B) C :ACB To represent (A C) (B C):ACB 10. 2Venn diagrams help us identify some useful formulas in set operations. To represent (A B) C :ACB To represent (A C) (B C):ACBTo prove (A B) C = (A C) (B C) in a rigorous manner, should useformal mathematical logic. 11. 3A few remarks about set notations If a set has nitely many elements , use the listing method to express the setwrite down all the elementsenclosed the elements by braces 12. 3A few remarks about set notations If a set has nitely many elements , use the listing method to express the setwrite down all the elementsenclosed the elements by bracesExample {1, 3, 5, 7, 9} 13. 3A few remarks about set notations If a set has nitely many elements , use the listing method to express the setwrite down all the elementsenclosed the elements by bracesExample {1, 3, 5, 7, 9} Because intervals have innitely many elements , we cant use listing methodto express intervals. We introduce new notations: 14. 3A few remarks about set notations If a set has nitely many elements , use the listing method to express the setwrite down all the elementsenclosed the elements by bracesExample {1, 3, 5, 7, 9} Because intervals have innitely many elements , we cant use listing methodto express intervals. We introduce new notations:square bracket means endpoint includedround bracket means endpoint excluded 15. 3A few remarks about set notations If a set has nitely many elements , use the listing method to express the setwrite down all the elementsenclosed the elements by bracesExample {1, 3, 5, 7, 9} Because intervals have innitely many elements , we cant use listing methodto express intervals. We introduce new notations:square bracket means endpoint includedround bracket means endpoint excludedExample [7, 11],(2, 5] 16. 3A few remarks about set notations If a set has nitely many elements , use the listing method to express the setwrite down all the elementsenclosed the elements by bracesExample {1, 3, 5, 7, 9} Because intervals have innitely many elements , we cant use listing methodto express intervals. We introduce new notations:square bracket means endpoint includedround bracket means endpoint excludedExample [7, 11],(2, 5]Note The interval [7, 11] contains ALL numbers between 7 and 11 (includingintegers, rational numbers, irrational numbers) 17. 3A few remarks about set notations If a set has nitely many elements , use the listing method to express the setwrite down all the elementsenclosed the elements by bracesExample {1, 3, 5, 7, 9} Because intervals have innitely many elements , we cant use listing methodto express intervals. We introduce new notations:square bracket means endpoint includedround bracket means endpoint excludedExample [7, 11],(2, 5]Note The interval [7, 11] contains ALL numbers between 7 and 11 (includingintegers, rational numbers, irrational numbers)For example, 10 [7, 11] 9.123 [7, 11] 50 [7, 11] 18. 4InequalitiesTo solve an inequality (or inequalities) in an unknown x means to nd all realnumbers x such that the inequality is satised .The set of all such x is called the solution set to the inequality. 19. 4InequalitiesTo solve an inequality (or inequalities) in an unknown x means to nd all realnumbers x such that the inequality is satised .The set of all such x is called the solution set to the inequality.Polynomial inequalities an xn + an1 xn1 + + a1 x + a0 < 0 ( or > 0, or 0, or 0)where n 1 and an 0. 20. 4InequalitiesTo solve an inequality (or inequalities) in an unknown x means to nd all realnumbers x such that the inequality is satised .The set of all such x is called the solution set to the inequality.Polynomial inequalities an xn + an1 xn1 + + a1 x + a0 < 0 ( or > 0, or 0, or 0)where n 1 and an 0. (1) n = 1Linear inequalities (2) n = 2Quadratic inequalities (3) n 3Higher degree inequalities 21. 5Example Find the solution set to the following compound inequality: 1 3 2x 9 22. 5Example Find the solution set to the following compound inequality: 1 3 2x 9Solution The inequality means 1 3 2x and 3 2x 9. 23. 5Example Find the solution set to the following compound inequality: 1 3 2x 9Solution The inequality means 1 3 2x and 3 2x 9.Solving separately:1 3 2x 24. 5Example Find the solution set to the following compound inequality: 1 3 2x 9Solution The inequality means 1 3 2x and 3 2x 9.Solving separately:1 3 2x2x 3 1 25. 5Example Find the solution set to the following compound inequality: 1 3 2x 9Solution The inequality means 1 3 2x and 3 2x 9.Solving separately:1 3 2x2x 3 1 x 1 26. 5Example Find the solution set to the following compound inequality: 1 3 2x 9Solution The inequality means 1 3 2x and 3 2x 9.Solving separately:1 3 2x 3 2x 92x 3 1 x 1 27. 5Example Find the solution set to the following compound inequality: 1 3 2x 9Solution The inequality means 1 3 2x and 3 2x 9.Solving separately:1 3 2x 3 2x 92x 3 13 9 2x x 1 28. 5Example Find the solution set to the following compound inequality: 1 3 2x 9Solution The inequality means 1 3 2x and 3 2x 9.Solving separately:1 3 2x 3 2x 92x 3 13 9 2x x 13 x 29. 5Example Find the solution set to the following compound inequality:1 3 2x 9Solution The inequality means 1 3 2x and 3 2x 9.Solving separately: 1 3 2x 3 2x 92x 3 1 3 9 2xx 13 x Solution set = {x R : x 1 and 3 x} 30. 5Example Find the solution set to the following compound inequality:1 3 2x 9Solution The inequality means 1 3 2x and 3 2x 9.Solving separately: 1 3 2x 3 2x 92x 3 1 3 9 2xx 13 x Solution set = {x R : x 1 and 3 x}= {x R : 3 x 1} 31. 5Example Find the solution set to the following compound inequality:1 3 2x 9Solution The inequality means 1 3 2x and 3 2x 9.Solving separately: 1 3 2x 3 2x 92x 3 1 3 9 2xx 13 x Solution set = {x R : x 1 and 3 x}= {x R : 3 x 1}= [3, 1] 32. 5Example Find the solution set to the following compound inequality:1 3 2x 9Solution The inequality means 1 3 2x and 3 2x 9.Solving separately: 1 3 2x 3 2x 92x 3 1 3 9 2xx 13 x Solution set = {x R : x 1 and 3 x}= {x R : 3 x 1}= [3, 1]Be careful1 3 2x1 3 2x 33. 5Example Find the solution set to the following compound inequality:1 3 2x 9Solution The inequality means 1 3 2x and 3 2x 9.Solving separately: 1 3 2x 3 2x 92x 3 1 3 9 2xx 13 x Solution set = {x R : x 1 and 3 x}= {x R : 3 x 1}= [3, 1]Be careful1 3 2x1 3 2x2 x2 34. 6Example Find the solution set to the following:2x + 1 < 3 and 3x + 10 < 4 35. 6Example Find the solution set to the following:2x + 1 < 3 and 3x + 10 < 4Solution Solve separately:2x + 1 < 3 2x < 2 x < 1 36. 6Example Find the solution set to the following:2x + 1 < 3 and 3x + 10 < 4Solution Solve separately:2x + 1 < 33x + 10 < 4 2x < 23x < 6 x < 1x < 2 37. 6Example Find the solution set to the following:2x + 1 < 3 and 3x + 10 < 4Solution Solve separately:2x + 1 < 33x + 10 < 4 2x < 23x < 6 x < 1x < 2 Solution set = {x R : x < 1 and x < 2} 38. 6Example Find the solution set to the following:2x + 1 < 3 and 3x + 10 < 4Solution Solve separately:2x + 1 < 33x + 10 < 4 2x < 23x < 6 x < 1x < 2 Solution set = {x R : x < 1 and x < 2}= {x R : x < 2} 39. 6Example Find the solution set to the following:2x + 1 < 3 and 3x + 10 < 4Solution Solve separately:2x + 1 < 33x + 10 < 4 2x < 23x < 6 x < 1x < 2 Solution set = {x R : x < 1 and x < 2}= {x R : x < 2}= (, 2) 40. 7Example Find the solution set to the following:2x + 1 > 9 and 3x + 4 < 10 41. 7Example Find the solution set to the following:2x + 1 > 9 and 3x + 4 < 10Solution Solve separately:2x + 1 > 9 2x > 8 x > 4 42. 7Example Find the solution set to the following:2x + 1 > 9 and 3x + 4 < 10Solution Solve separately:2x + 1 > 93x + 4 < 10 2x > 8 3x < 6 x > 4 x < 2 43. 7Example Find the solution set to the following:2x + 1 > 9 and 3x + 4 < 10Solution Solve separately:2x + 1 > 93x + 4 < 10 2x > 8 3x < 6 x > 4 x < 2 Solution set = {x R : x > 4 and x < 2} 44. 7Example Find the solution set to the following:2x + 1 > 9 and 3x + 4 < 10Solution Solve separately:2x + 1 > 93x + 4 < 10 2x > 8 3x < 6 x > 4 x < 2 Solution set = {x R : x > 4 and x < 2}= 45. 8Wording(1) Find the solution(s) to the inequality 2x 1 > 0.(2) Find the solution set to the inequality 2x 1 > 0. 46. 8Wording(1) Find the solution(s) to the inequality 2x 1 > 0.(2) Find the solution set to the inequality 2x 1 > 0.Answer 1(1) Solutions x> 2 47. 8Wording(1) Find the solution(s) to the inequality 2