Math 51/COEN 19

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, ar2 , ar3, …, arn, …

• A recurrence relation defines the nth term of a sequence in terms of some of the previous terms.

• A formula for an that you can write down without …, ∑, or previous terms is a closed formula.

• Solving a recurrence relation means finding a closed formula for the nth term.

Solve the recurrence relation:

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

sum=0for i=1 to 5

for j=1 to Isum = sum+i+j

print(sum)

Cardinality of sets

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|

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”

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

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 r1, r2, r3, … then we have shown the set is countable

Do you think the following sets are countable?

• All integers• All rationals • All reals