1. Chapter 1 Section 1 - Slide 1 Copyright © 2009 Pearson Education, Inc. AND

2. Chapter 1 Section 1 - Slide 2 Copyright © 2009 Pearson Education, Inc. Chapter 1.1 Chapter 2.1, 2.2 Critical Thinking Skills Set Concepts and Subsets

3. Chapter 1 Section 1 - Slide 3 Copyright © 2009 Pearson Education, Inc. WHAT YOU WILL LEARN Inductive and deductive reasoning processes Methods to indicate sets, equal sets, and equivalent sets Subsets and proper subsets

4. Chapter 1 Section 1 - Slide 4 Copyright © 2009 Pearson Education, Inc. Section 1.1 Inductive Reasoning

5. Chapter 1 Section 1 - Slide 5 Copyright © 2009 Pearson Education, Inc. Natural Numbers The set of natural numbers is also called the set of counting numbers. The three dots, called an ellipsis, mean that 8 is not the last number but that the numbers continue in the same manner.

6. Chapter 1 Section 1 - Slide 6 Copyright © 2009 Pearson Education, Inc. Divisibility If a  b has a remainder of zero, then a is divisible by b. The even counting numbers are divisible by 2. They are 2, 4, 6, 8,…. The odd counting numbers are not divisible by 2. They are 1, 3, 5, 7,….

7. Chapter 1 Section 1 - Slide 7 Copyright © 2009 Pearson Education, Inc. Inductive Reasoning The process of reasoning to a general conclusion through observations of specific cases. Also called induction. Often used by mathematicians and scientists to predict answers to complicated problems.

8. Chapter 1 Section 1 - Slide 8 Copyright © 2009 Pearson Education, Inc. Scientific Method Inductive reasoning is a part of the scientific method. When we make a prediction based on specific observations, it is called a conjecture.

9. Chapter 1 Section 1 - Slide 9 Copyright © 2009 Pearson Education, Inc. Counterexample In testing a conjecture, if a special case is found that satisfies the conditions of the conjecture but produces a different result, that case is called a counterexample.  Only one exception is necessary to prove a conjecture false.  If a counterexample cannot be found, the conjecture is neither proven nor disproven.

10. Chapter 1 Section 1 - Slide 10 Copyright © 2009 Pearson Education, Inc. Deductive Reasoning A second type of reasoning process. Also called deduction. Deductive reasoning is the process of reasoning to a specific conclusion from a general statement.

11. Chapter 1 Section 1 - Slide 11 Copyright © 2009 Pearson Education, Inc. Example: Inductive Reasoning Use inductive reasoning to predict the next three numbers in the pattern (or sequence). 7, 11, 15, 19, 23, 27, 31,… Solution: We can see that four is added to each term to get the following term. 31 + 4 = 35, 35 + 4 = 39, 39 + 4 = 43 Therefore, the next three numbers in the sequence are 35, 39, and 43.

12. Chapter 1 Section 1 - Slide 12 Copyright © 2009 Pearson Education, Inc. Section 2.1 Set Concepts

13. Chapter 1 Section 1 - Slide 13 Copyright © 2009 Pearson Education, Inc. Set A collection of objects, which are called elements or members of the set. Listing the elements of a set inside a pair of braces, { }, is called roster form. The symbol, read “is an element of,” is used to indicate membership in a set. The symbol means “is not an element of.”

14. Chapter 1 Section 1 - Slide 14 Copyright © 2009 Pearson Education, Inc. Well-defined Set A set which has no question about what elements should be included. Its elements can be clearly determined. No opinion is associated its the members.

15. Chapter 1 Section 1 - Slide 15 Copyright © 2009 Pearson Education, Inc. Roster Form This is the form of the set where the elements are all listed, separated by commas. Example: Set A is the set of all natural numbers less than or equal to 25. Solution: A = {1, 2, 3, 4, 5,…, 25} The 25 after the ellipsis indicates that the elements continue up to and including the number 25.

16. Chapter 1 Section 1 - Slide 16 Copyright © 2009 Pearson Education, Inc. Set-Builder (or Set-Generator) Notation A formal statement that describes the members of a set is written between the braces. A variable may represent any one of the members of the set. Example: Write set B = {2, 4, 6, 8, 10} in set- builder notation. Solution: The set of all x such that x is a natural number and x is an even number 10.

17. Chapter 1 Section 1 - Slide 17 Copyright © 2009 Pearson Education, Inc. Finite Set A set that contains no elements or the number of elements in the set is a natural number. Example: Set S = {2, 3, 4, 5, 6, 7} is a finite set because the number of elements in the set is 6, and 6 is a natural number.

18. Chapter 1 Section 1 - Slide 18 Copyright © 2009 Pearson Education, Inc. Infinite Set An infinite set is a set where the number of elements is not or a natural number; that is, you cannot count the number of elements. The set of natural numbers is an example of an infinite set because it continues to increase forever without stopping, making it impossible to count its members.

19. Chapter 1 Section 1 - Slide 19 Copyright © 2009 Pearson Education, Inc. Equal sets have the exact same elements in them, regardless of their order. Symbol: A = B Example: { 1, 5, 7 } = { 5, 7, 1 } Equal Sets

20. Chapter 1 Section 1 - Slide 20 Copyright © 2009 Pearson Education, Inc. Cardinal Number The number of elements in set A is its cardinal number. Symbol: n(A) Example:A = { 1, 5, 7, 10 } n(A) = 4

21. Chapter 1 Section 1 - Slide 21 Copyright © 2009 Pearson Education, Inc. Equivalent Sets Equivalent sets have the same number of elements in them. Symbol: n(A) = n(B) Example:A = { 1, 5, 7 }, B = { 2, 3, 4 } n(A) = n(B) = 3 So A is equivalent to B.

22. Chapter 1 Section 1 - Slide 22 Copyright © 2009 Pearson Education, Inc. Empty (or Null) Set The null set (or empty set ) contains absolutely NO elements. Symbol:

23. Chapter 1 Section 1 - Slide 23 Copyright © 2009 Pearson Education, Inc. Universal Set The universal set contains all of the possible elements which could be discussed in a particular problem. Symbol: U

24. Chapter 1 Section 1 - Slide 24 Copyright © 2009 Pearson Education, Inc. Section 2.2 Subsets

25. Chapter 1 Section 1 - Slide 25 Copyright © 2009 Pearson Education, Inc. Subsets A set is a subset of a given set if and only if all elements of the subset are also elements of the given set. Symbol: To show that set A is not a subset of set B, one must find at least one element of set A that is not an element of set B. The symbol for “not a subset of” is.

26. Chapter 1 Section 1 - Slide 26 Copyright © 2009 Pearson Education, Inc. Example: Determine whether set A is a subset of set B. A = { 3, 5, 6, 8 } B = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} Solution: All of the elements of set A are contained in set B, so Determining Subsets A B .

27. Chapter 1 Section 1 - Slide 27 Copyright © 2009 Pearson Education, Inc. Proper Subset All subsets are proper subsets except the subset containing all of the given elements, that is, the given set must contain one element not in the subset (the two sets cannot be equal). Symbol:

28. Chapter 1 Section 1 - Slide 28 Copyright © 2009 Pearson Education, Inc. Determining Proper Subsets Example: Determine whether set A is a proper subset of set B. A = { dog, cat } B = { dog, cat, bird, fish } Solution: All the elements of set A are contained in set B, and sets A and B are not equal, therefore A B.

29. Chapter 1 Section 1 - Slide 29 Copyright © 2009 Pearson Education, Inc. Determining Proper Subsets (continued) Example: Determine whether set A is a proper subset of set B. A = { dog, bird, fish, cat } B = { dog, cat, bird, fish } Solution: All the elements of set A are contained in set B, but sets A and B are equal, therefore A B.

30. Chapter 1 Section 1 - Slide 30 Copyright © 2009 Pearson Education, Inc. Number of Distinct Subsets The number of distinct subsets of a finite set A is 2 n, where n is the number of elements in set A. Example: Determine the number of distinct subsets for the given set { t, a, p, e }. List all the distinct subsets for the given set: { t, a, p, e }.

31. Chapter 1 Section 1 - Slide 31 Copyright © 2009 Pearson Education, Inc. Solution: Since there are 4 elements in the given set, the number of distinct subsets is 2 4 = 2 2 2 2 = 16 subsets. {t,a,p,e}, {t,a,p}, {t,a,e}, {t,p,e}, {a,p,e}, {t,a}, {t,p}, {t,e}, {a,p}, {a,e}, {p,e}, {t}, {a}, {p}, {e}, { } Number of Distinct Subsets continued