1. Cardinality of a Set “The number of elements in a set.” Let A be a set. a.If A =  (the empty set), then the cardinality of A is 0. b. If A has exactly n elements, n a natural number, then the cardinality of A is n. The set A is a finite set. c. Otherwise, A is an infinite set.

2. Notation The cardinality of a set A is denoted by | A |. a.If A = , then | A |= 0. b.If A has exactly n elements, then | A | = n. c. If A is an infinite set, then | A | = .

3. Examples: A = {2, 3, 5, 7, 11, 13, 17, 19}; | A | = 8 A = N (natural numbers); | N | =  A = Q (rational numbers); | Q | =  A = {2n | n is an integer}; | A | =  (the set of even integers)

4. DEFINITION: Let A and B be sets. Then, |A| = |B| if and only if there is a one-to-one correspondence between the elements of A and the elements of B. Examples: 1. A = {1, 2, 3, 4, 5} B = {a, e, i, o, u} 1  a, 2  e, 3  i, 4  o, 5  u; |B| = 5

5. 2. A = N (the natural numbers) B = {2n | n is a natural number} (the even natural numbers) n  2n is a one-to one correspondence between A and B. Therefore, |A| = |B|; |B| = . 3. A = N (the natural numbers) C = {2n  1 | n is a natural number} (the odd natural numbers) n  2n  1 is a one-to one correspondence between A and C. Therefore, |A| = |C|; |C| = .

6. Countable Sets DEFINITIONS: 1. A set S is finite if there is a one-to-one correspondence between it and the set {1, 2, 3,..., n} for some natural number n. 2. A set S is countably infinite if there is a one-to-one correspondence between it and the natural numbers N.

7. 3.A set S is countable if it is either finite or countably infinite. 4.A set S is uncountable if it is not countable.

8. Examples: 1. A = {1, 2, 3, 4, 5, 6, 7},  = {a, b, c, d,... x, y, z} are finite sets; |A| = 7, |  | = 26. 2. N (the natural numbers), Z (the integers), and Q (the rational numbers) are countably infnite sets; that is, |Q| = |Z| = |N|.

9. 3. I (the irrational numbers) and   (the real numbers) are uncountable sets; that is |I| > |N| and |  | > |N|.

10. Some Facts: 1.A set S is finite if and only if for any proper subset A  S, |A| < |S|; that is, “proper subsets of a finite set have fewer elements.” 2.Suppose that A and B are infinite sets and A  B. If B is countably infinite then A is countably infinite and |A| = |B|.

11. 3. Every subset of a countable set is countable. 4.If A and B are countable sets, then A  B is a countable set.

12. Irrational Numbers, Real Numbers Irrational numbers: “points on the real line that are not rational points”; decimals that are neither repeating nor terminating. Real numbers: “rationals”  “irrationals”

13. is a real number:

14. is not a rational number, i.e., is an irrational number. Proof: Suppose is a rational number. Then...

15. Other examples of irrational numbers: Square roots of rational numbers that are not perfect squares. Cube roots of rational numbers that are not perfect cubes. And so on.   3.14159, e  2.7182182845

16. Algebraic numbers – roots of polynomials with integer coefficients. Transcendental numbers – irrational numbers that are not algebraic.

17. THEOREM: The real numbers are uncountable! Proof: Consider the real numbers on the interval [0,1]. Suppose they are countable. Then... Arrive at a contradiction. COROLLARY: The irrational numbers are uncountable. Proof: Real numbers: “rationals”  “irrationals”

18. The Real Line

19. Absolute Value DEFINITION: Let a be a real number. The absolute value of a, denoted |a|, is given by Geometric interpretation: |a| is the distance on the real number line from the point a to the origin 0.

20. |5|=5, |  3|=3

21. Absolute value inequalities Find the real numbers x that satisfy: 1. |x| < 3 2. |x|  2 3. |x  3|  4 4. |x + 2| > 5 5. |2x  3| < 5

22. Answers: 1.(  3,3);  3 < x < 3 2.( ,  2]  [2,  ); x   2 or x  2 3.[  1,7];  1  x  7 4. ( ,  7)  (3,  ); x 3 5. (  1,4);  1 < x < 4