Corneille meaning to win without risk is a triumph

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Unformatted text preview: language Like French: “A vaincre sans péril, on triomphe sans gloire.” – Corneille Meaning: “To win without risk is a triumph without glory.” Like C++: #include <iostream> using namespace std; void main() { cout << "Hello World!" << endl; cout << "Welcome to C++ Programming" << endl; } Words – Probability models, random variables, expectation ... Grammar – Axioms, theorems, principles You have learned it well when you know how to use it M. Chen ([email protected]) ENGG2430 lecture 1 5 / 16 Set Why do we care: Probability use sets and set operations extensively A set is a collection of objects, which are the elements of the set. If S is a set and x is an element of S , we write x ∈ S Defining set S = {x1 , x2 , . . . , xn } S = {x | x satisfies condition P } Example: the set of possible outcomes of a coin toss is {Head, Tail} Subset: a set S is a subset of another set T , written as S ⊆ T , if every element of S is also an element of T , e.g., S = {all IE students} ⊆ T = {all CU students} Two sets S and T are equal, written as S = T , if S ⊆ T and T ⊆ S (a useful proof logic) M. Chen ([email protected]) ENGG2430 lecture 1 6 / 16 Set Operations The universal set Ω and the empty set 0 / Complement of a set S S c = {x ∈ Ω|x ∈ S } / Union of two sets S and T S ∪ T = {x |x ∈ S or x ∈ T } Intersection of two sets S and T S ∩ T = {x |x ∈ S and x ∈ T } M. Chen ([email protected]) ENGG2430 lecture 1 7 / 16 Set Operations – Examples Define the universal set Ω = {x | x ∈ (−∞, ∞)} Define sets S = {x |x > 1}, T = {x |x < −2} Questions: / S c = {x ∈ Ω|x ∈ S } =? S ∪ T = {x |x ∈ S or x ∈ T } =? S ∩ T = {x |x ∈ S and x ∈ T } =? S ...
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This document was uploaded on 03/31/2014.

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