1. Slide 7- 1 Copyright © 2012 Pearson Education, Inc.

2. 4.3 Intersections, Unions, and Compound Inequalities ■ Intersections of Sets and Conjunctions of Sentences ■ Unions of Sets and Disjunctions of Sentences ■ Interval Notation and Domains

3. Slide 4- 3 Copyright © 2012 Pearson Education, Inc. Intersection of Sets and Conjunctions of Sentences The intersection of two sets A and B is the set of all elements that are common to both A and B. We denote the intersection of sets A and B as AB

4. Slide 4- 4 Copyright © 2012 Pearson Education, Inc. Example Solution Find the intersection: The letters a and e are common to both sets, so the intersection is {a, e}.

5. Slide 4- 5 Copyright © 2012 Pearson Education, Inc. When two or more sentences are joined by the word and to make a compound sentence, the new sentence is called a conjunction of the sentences. Example –1 < x and x < 3. A number is a solution of a conjunction if it is a solution of both of the separate parts.

6. Slide 4- 6 Copyright © 2012 Pearson Education, Inc. Note that the solution set of a conjunction is the intersection of the solution sets of the individual sentences.

7. Slide 4- 7 Copyright © 2012 Pearson Education, Inc. Example Solution Solve and graph:

8. Slide 4- 8 Copyright © 2012 Pearson Education, Inc. Note that for a < b, and, equivalently, So 3 < 2x +1 < 7 can be solved as 3 < 2x +1 and 2x + 1 < 7 a < x and x < b can be abbreviated a < x < b; b > x and x > a can be abbreviated b > x > a.

9. Slide 4- 9 Copyright © 2012 Pearson Education, Inc. Mathematical Use of the Word “and” The word “and” corresponds to “intersection” and to the symbol Any solution of a conjunction must make each part of the conjunction true.

10. Slide 4- 10 Copyright © 2012 Pearson Education, Inc. Sometimes there is no way to solve both parts of a conjunction at once. AB In this situation, A and B are said to be disjoint.

11. Slide 4- 11 Copyright © 2012 Pearson Education, Inc. Example Solution Solve and graph: Since no number is greater than 5 and less than 1, the solution set is the empty set 0

12. Slide 4- 12 Copyright © 2012 Pearson Education, Inc. The union of two sets A and B is the collection of elements belonging to A and/or B. We denote the union of sets A and B by Unions of Sets and Disjunctions of Sentences AB

13. Slide 4- 13 Copyright © 2012 Pearson Education, Inc. Example Solution Find the union: The letters in either or both sets are a, b, c, d, e, i, o and u, so the union is

14. Slide 4- 14 Copyright © 2012 Pearson Education, Inc. When two or more sentences are joined by the word or to make a compound sentence, the new sentence is called a disjunction of the sentences. Example x 8 A number is a solution of a disjunction if it is a solution of at least one of the separate parts.

15. Slide 4- 15 Copyright © 2012 Pearson Education, Inc. Note that the solution set of a disjunction is the union of the solution sets of the individual sentences.

16. Slide 4- 16 Copyright © 2012 Pearson Education, Inc. Mathematical Use of the Word “or” The word “or” corresponds to “union” and to the symbol For a number to be a solution of a disjunction, it must be in at least one of the solution sets of the individual sentences.

17. Slide 4- 17 Copyright © 2012 Pearson Education, Inc. Example Solution Solve and graph:

18. Slide 4- 18 Copyright © 2012 Pearson Education, Inc. Interval Notation and Domains We saw earlier that if, then the domain We can now represent such a set using interval notation: 7

19. Slide 4- 19 Copyright © 2012 Pearson Education, Inc. Domain of the Sum, Difference, or Product of Functions The domain of the sum, the difference, or the product of the functions f and g is the intersection of the domains of f and g.