Definition: An Alphabet is finite to non-empty set of symbols ( letters)
A word w (over ) is finite set of symbols from
* (sigmaster) is the set of all words = infinite
A word is any finite sequence of symbols from / over Alphabet
Empty word, null word,

What is a set? A collection of elements
Properties of sets: in a set order does not matter no repeated elements
Capital letters stand for a set
A= cfw_2,3,4,5 is the same as A=cfw_5,4,3,2
B= cfw_1,2,3,4 B is the set of integers from one to four inclusive

Relation is any sublet of ordered pairs of a Cartesian product of two sets
Reflexive all x E S are in relation and all related to themselves
Anti-reflexive nothing is related to its self
Symmetric if there is any (x,y) there is also a (y,x)
Anti-Symmetric

Induction: a general method for proving something
a) State the base case, prove its true
b) State the induction step , show its true
If P(k) is true for some integer K then P(k+1) is also true
P(n) = 2n < n! n 4
24 < 4! -> 16 < 24 state base case prove it

Probability: each prob 0 x 1 adding them all together = 1
Simple events list of all possible events
Sample space is the set of all simple events
Complex events = sets of simple events
P(A U B) = P(A) + P(B) P(A B )
Easier if events are equally likely ( th

Adjacency matrix if there is an edge put a one other wise put a zero
Reachability matrix if there is a path between two nodes put a one
Finite and infinite sets: all infinite sets have a one to one correspondence except
the power set of an infinite set an

S is the set of real numbers which are greater then equal to zero
X R y -> |_x_| = |_y_|
Is this relation R reflexive? Yes, every x will always be realted to itself
because the floor function on |_4_| is the same as |_4_| all the time a
number will always

Pick a 3 letter word at random from all three letter words over alphabet = cfw_a,b,c
What is the sample space for the above expirment?
27 possibilities (33)
aaa - bbb ccc abc cba - bca bac bba bbc cca ccb bcb cbb
abb acc aab aac aca aba baa bab cac bcc c

Truth tables
In a truth table with four variable p,q,r,s = 24 = each event has probability of 1/16
A is the event that all four variable in that row have the same truth value
A = cfw_TTTT, FFFF
P(A) =2/16 = 1/8
B is the event that the second and third va

Use relation R on a set S to describe the digraph G
A set of ordered pairs where the first of the ordered pairs is a node from digraph G
and the second of the ordered pair is another node from digraph G where there is
an edge from the 1st node to the seco