CS723 - Probability and Stochastic Processes Lecture 02

In Previous Lecture Unpredictable World Zero/Partial Knowledge Complicated Model Mathematical Analysis

Venn Diagram U A B

Venn Diagram A = { 1, 2, 3 } U 1 2 3 A

Ac = { 7, a, b, c, lion, elephant, stray dog} Complement of a Set A = { 1, 2, 3, 4, 5 } Ac = { 7, a, b, c, lion, elephant, stray dog} ? Ac = U – A = { 6, 7, 8, 9, 10 } where U = { 1, 2, 3, … , 10 }

Venn Diagram Complement of a set U 7 8 6 10 4 1 2 5 3 A 9

Power Set A = { 1, 2 } Pow (A) = { { }, {1}, {2}, {1, 2} } A = { 1, 2, 3 } = { a2 , a1 , a0 } Pow (B) = { b: b = (b2b1b0)2 0 ≤ b < 7, bi = 1 ai is in the subset }

Cardinality of Power Set P = Cardinality of set A, P < ∞ Q = 2P = Cardinality of Pow (A) , Q < ∞ A = { 1, 2, 3, … } = Natural Numbers Q = 2P = ? Pow (A) = { b: b = (0.b1b2b3…)2 0 ≤ b < 1, bi = 1 Number i is in the subset }

Power Set of Infinite Set P = Cardinality of infinite set e.g. N P = ∞, the countable type Q = 2P = Cardinality of Pow (N) Q = ∞, the uncountable type R = 2N Cardinality of Pow (R) = 2R = ? = ∞ ?

Cartesian Product Ordered collection of elements from sets A = { x, y, z } B = { 1, 2, 3 } A x B = { (x,1), (x,2), (x,3), (y,1), (y,2), (y,3), (z,1), (z,2), (z,3) } Cardinality of Cartesian product If | A | = P < ∞ and | B | = Q < ∞ , then | A x B | = PQ < ∞

Cartesian Product If A = Z and B = Z , then A x B = Z2 = { (x1 , x2), x1 in Z, x2 in Z} | A | = | B | = | A x B | = Countably infinite Use If A = R and B = R , then A x B = R2 = { (x1 , x2), x1 in R, x2 in R} | A | = | B | = Uncountably infinite | A x B | = ? R x R x R = (R x R) x R = R x (R x R) = R3

Sets and Vectors Sets : Unordered collection of objects Vectors : Ordered collection of objects A = { x, y } B = { 1, 2 } A x B = { (x ,1) , (x , 2) , (y ,1) , (y , 2)} = {(y ,1) , (x , 2) , (y , 2) , (x ,1)} (x ,1) (A x B) but (1, x) (A x B)

Sets and Vectors A = set of your uncles = { u1, u2, u3, u4 , u5 } = { u4, u5, u1, u2 , u3 } = { u1, u4, u3, u5 , u2 } b = vector of your male ancestors = ( f, gf, ggf, gggf , … , Adam ) ≠ ( f, gggf, Adam,ggf, … , gf ) B = set of your male ancestors = { f, gggf, Adam,ggf, … , gf }

Relation Subset of a Cartesian Product A x B All pairs that satisfy some criterion Domain = Subset of A, set of starting points in pairs in a relation Range = Subset of B, set of terminating points is pair in a relation All possible pairs formed from entries of Domain and Range are not required to be in a relation

Relations Relations in R2 r1 = { (x,y) s.t. -5 < x < 5 , 0 < x+y < 10 } r2 = { (x,y) s.t. -5 < x < 5 , x2 + y2 < 25 } r3 = { (x,y) s.t. 0 < x < 9 , sin(x) < 0.5 } Other relations in R2

Relations in daily life A = Set of all adult living human males B = Set of all adult living human females A x B = { (a,b), a is a male, b is a female } r1 = { (a,b), a has same religion as b } r2 = { (a,b), a lives in same city as b } r3 = { (a,b), a has met b, at least once } r4 = { (a,b), a wants to marry b } r5 = { (a,b), a can marry b, realistically } r6 = { (a,b), a is husband of b }

Function Special type of relation For every domain element, there is only one unique element of the range A functions is also called a mapping Functions in R2 , a subset of R x R f1 = { (x,y) 0 < x < 10 , y = 3x + 2 } also f1: R → R = x → 3x+2

Function Graphs of functions in R2

Examples of Functions Function of two variables ( R x R ) x R Surface over plane ( R x R ) → R Complex valued function R x ( R x R ) Complex exponential R → ( R x R )

Functions in daily life A = Set of all human males B = Set of all human females A x B = { (a,b), a is a male, b is a female } f1 = { (a,b), a is a son of b } r = { (a,b), a is husband of b } f3 = { (b,a) (B x A), b is wife of a }