1. Math 51/COEN 19

2. Sequences and Summations - vocab An arithmetic progression is a sequence of the form a, a+d, a+2d, …, a+nd, … with fixed a, d in R and varying n in Z >=0 A geometric progression is a sequence of the form a, ar, ar 2, ar 3, …, ar n, … A recurrence relation defines the nth term of a sequence in terms of some of the previous terms. A formula for a n that you can write down without …, ∑, or previous terms is a closed formula. Solving a recurrence relation means finding a closed formula for the n th term.

4. Solve the recurrence relation:

8. I expect you to know off the top of your head:

9. sum=0 for i=1 to 5 for j=1 to I sum = sum+i+j print(sum)

10. Cardinality of sets

11. Cardinality vocab The cardinality of a finite set is the number of elements in the set. The sets A and B have the same cardinality iff there is a bijection between A and B. When A and B have the same cardinality we write |A|=|B| If there is a one-to-one function from A to B, the cardinality of A is less than or the same as the cardinality of B and we write |A|<=|B| Moreover, when |A|<=|B| and A and B have different cardinality, we say that the cardinality of A is less than the cardinality of B and we write |A|<|B|

12. More Cardinality Vocab A set that is either finite or has the same cardinality as the set of positive integers is called countable. A set that is not countable is called uncountable. When an infinite set S is countable, we denote the cardinality of S by 0 We write |S| = 0 and say that S has cardinality “aleph null”

13. More Cardinality Vocab A set that is either finite or has the same cardinality as the set of positive integers is called countable. A set that is not countable is called uncountable. From a practical viewpoint, to show a set S is countable, we find a bijection between the positive integers and S. The only cardinalities we care about (for this class) for infinite sets are countable vs uncountable

14. More Cardinality Vocab A set that is either finite or has the same cardinality as the set of positive integers is called countable. A set that is not countable is called uncountable. From a practical viewpoint, to show a set S is countable, we find a bijection between the positive integers and S. If we can list all the members of a set as a sequence r 1, r 2, r 3, … then we have shown the set is countable

15. Do you think the following sets are countable? All integers All rationals All reals