*This preview shows
pages
1–3. Sign up
to
view the full content.*

This
** preview**
has intentionally

**sections.**

*blurred***to view the full version.**

*Sign up*
**Unformatted text preview: **MA100, Fall 2010, PART 1: Precalculus [1] The Set of Real Numbers 1. Sets in general. Notations A set is a collection of objects, called the elements or members of the set. The elements of a set are listed between braces (curly brackets). The elements of a set are listed only once and the order in which these elements are listed is irrelevant. Example: { red, blue, green } is a set formed by the colors red, blue and green. Example: { 22 , 23 , 24 , 25 } is a set formed by the numbers 22 , 23 , 24 , 25. Notation : Sets are usually denoted by capital letters. Example: A = { b,d,a,f,g,h } is a set denoted by A and it is formed by the letters b,d,a,f,g,h . Example: X = { 22 , 23 , 24 , 25 } is a set denoted by X and it is formed by the numbers 22 , 23 , 24 , 25. Notation : The elements of a set can be either listed one by one, or described by a property that characterizes them. Example: The set X = { 22 , 23 , 24 , 25 } can be equivalently written as X = { the integers x with the property that x is greater than 21 and x is less than 26 } or X = { the integers x such that 21 < x < 26 } or X = { the integers x : 21 < x < 26 } Note that above, : is a short-hand notation for the words such that . (Recall that , 21 < x is a short-hand notation for all numbers x that are greater than 21. Similarly, x < 26 is a short-hand notation for all numbers x that are less than 26.) Notation : The membership of an element to a set is denoted by the symbol . The non- membership is denoted by / . Example: Consider the set A = { b,d,a,f,g,h } . g A which reads g is an element of the set A or g is in A 2 / A which reads 2 is not element of the set A or 2 is not in A m / A which reads m is not element of the set A or m is not in A Definition : A set that contains no element is called the empty or the void set and it is denoted by the greek letter (which is read phi). Definition : Two sets are equal (or the same) if they consists of the same elements. In this case, we use the symbol =. If two sets do not contain the same elements, then they are not equal and we denote this by 6 = . Example: Consider the sets A = { b,d,a,f,g,h } and B = { b,d,a,f,g,h } . Obviously, A = B. 1 Example: Consider the sets A = { b,d,a,f,g,h } and B = { a,b,d,f,g,h } . Here also A = B, since A and B contain the same elements (recall that, when listing the elements of a set, the order doesnt matter). Example: Consider the sets A = { b,d,a,f,g,h } and C = { b,d,f,g,h, 7 } . Then A 6 = C. Definition : Let A and B be two sets. If all the elements of A are also elements of B , then we say that A is a subset of B and use the notation A B....

View
Full
Document