This preview shows pages 1–12. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Stat 5101 Lecture Slides Deck 1 Charles J. Geyer School of Statistics University of Minnesota 1 Sets In mathematics, a set is a collection of objects thought of as one thing. The objects in the set are called its elements . The notation x ∈ S says that x is an element of the set S . The notation A ⊂ S says that the set A is a subset of the set S , that is, every element of A is an element of S . 2 Sets (cont.) Sets can be indicated by listing the elements in curly brackets { 1 , 2 , 3 , 4 } . Sets can collect anything, not just numbers { 1 , 2 ,π, cabbage , { , 1 , 2 }} One of the elements of this set is itself a set { , 1 , 2 } . Most of the sets we deal with are sets of numbers or vectors. 3 Sets (cont.) The empty set {} is the only set that has no elements. Like the number zero, it simplifies a lot of mathematics, but isn’t very interesting in itself. The empty set has its own special notation ∅ . 4 Sets (cont.) Some very important sets also get their own special notation. • N denotes the natural numbers { , 1 , 2 ,... } . • Z denotes the integers { ..., 2 , 1 , , 1 , 2 ,... } . • R denotes the real numbers. 5 Sets (cont.) Another notation for sets is the set builder notation { x ∈ S : some condition on x } denotes the set of elements of S that satisfy the specified con dition. For example, { x ∈ R : x > } is the set of positive real numbers. 6 Intervals Another important special kind of set is an interval . We use the notation ( a,b ) = { x ∈ R : a < x < b } (1) [ a,b ] = { x ∈ R : a ≤ x ≤ b } (2) ( a,b ] = { x ∈ R : a < x ≤ b } (3) [ a,b ) = { x ∈ R : a ≤ x < b } (4) which assumes a and b are real numbers such that a < b . (1) is called the open interval with endpoints a and b ; (2) is called the closed interval with endpoints a and b ; (3) and (4) are called halfopen intervals. 7 Intervals (cont.) We also use the notation ( a, ∞ ) = { x ∈ R : a < x } (5) [ a, ∞ ) = { x ∈ R : a ≤ x } (6) (∞ ,b ) = { x ∈ R : x < b } (7) (∞ ,b ] = { x ∈ R : x ≤ b } (8) (∞ , ∞ ) = R (9) which assumes a and b are real numbers. (5) and (7) are open intervals. (6) and (8) are closed intervals. (9) is both open and closed. 8 Functions A mathematical function is a rule that for each point in one set called the domain of the function gives a point in another set called the codomain of the function. Functions are also called maps or mappings or transformations . Functions are often denoted by single letters, such as f , in which case the rule maps points x in the domain to values f ( x ) in the codomain. f is a function, f ( x ) is the value of this function at the point x . 9 Functions (cont.) If X is the domain and Y the codomain of the function f , then to indicate this we write f : X → Y or X f→ Y 10 Functions (cont.) To define a function, we may give a formula f ( x ) = x 2 , x ∈ R ....
View
Full
Document
This note was uploaded on 02/07/2012 for the course STAT 5101 taught by Professor Staff during the Fall '02 term at Minnesota.
 Fall '02
 Staff
 Statistics

Click to edit the document details