s1 - Stat 5101 Lecture Slides Deck 1 Charles J. Geyer...

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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 half-open 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 ....
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This note was uploaded on 02/07/2012 for the course STAT 5101 taught by Professor Staff during the Fall '02 term at Minnesota.

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s1 - Stat 5101 Lecture Slides Deck 1 Charles J. Geyer...

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