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

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Stat 5101 Lecture Slides Deck 1 Charles J. Geyer School of Statistics University of Minnesota 1

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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 , { 0 , 1 , 2 }} One of the elements of this set is itself a set { 0 , 1 , 2 } . Most of the sets we deal with are sets of numbers or vectors. 3

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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 { 0 , 1 , 2 , . . . } . Z denotes the integers { . . . , - 2 , - 1 , 0 , 1 , 2 , . . . } . R denotes the real numbers. 5

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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 > 0 } 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

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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

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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 . Note that we indicate the domain in the formula. The same function can be indicated more simply by x 7→ x 2 , read x maps to x 2 .” This “maps to” notation does not indicate the domain, which must be indicated some other way.

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

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