sets_realnos_intervals

sets_realnos_intervals - Sets, Real Numbers and Intervals...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Sets, Real Numbers and Intervals Sets and set operations A set is any well-defined collection of (mathematical) objects. A set can be specified by listing its elements , or members, within curly brackets { } . . . . . = A { } , , , 1 2 3 4 and = B { } , , , ,- 4- 2 0 2 4 are two finite sets of integers or "whole numbers". The set of all integers is the infinite set = Z { } , , , , , , , , , , , , . . . . - 5- 4- 3- 2- 1 0 1 2 3 4 5 . . . . , where . . . . indicates that the listing continues forever. The order in which the elements of a set are listed is immaterial. Thus the set A above could equally well be specified by the equation = A { } , , , 3 1 4 2 . The important notion behind the concept of a set is membership . A set is determined only by knowing precisely which objects belong to the set. We write, 3 Z , or 3 Z , to mean 3 "is a member of", or "is an element of", the set of integers Z. A set can be specified by means of a general description of its elements, which is sufficiently precise to determine exactly which objects belong to the set. The sets A and B above could be defined as follows, where the symbol " | " should be read as "such that". = A { } , n | n Z and 1 n n 4 , that is, A is the set of all integers which are greater than or equal to 1 and less than or equal to 4. = B { } , 2 n | n Z and - 2 n n 2 that is, B is the set of all integers of the form 2 n , where n is greater than or equal to - 2 and less than or equal to 2. These are the even integers between - 4 and 4 inclusive. The set of all even integers is = E { } 2 n | n Z = { } , , , , , , , , , , , , . . . . - 10- 8- 6- 4- 2 0 2 4 6 8 10 . . . . , and the set of all odd integers is = O { }- 2 n 1 | n Z = { } , , , , , , , , , , , , . . . . - 9- 7- 5- 3- 1 1 3 5 7 9 11 . . . . . The set of perfect squares of integers is = S { } and n 2 | n Z n = { } , , , , , , , 0 1 4 9 16 25 36 . . . . . The intersection of two sets Given any two sets A and B , their intersection , denoted by A B , is the set of all elements which belong to both A and B , that is, it is the set of elements common to the two sets A an B . = A B { } and x | x A x B ________________ For example, if = A { } , , , 1 2 3 4 and = B { } , , , ,- 4- 2 0 2 4 then = A B { } , 2 4 . If , E O and S are the sets of even integers, odd integers and squares of integers, as above, then = E S { } , , , , 0 4 16 36 . . . . = { } and 4 n 2 | n Z n , and = O S { } , , , , 1 9 25 49 . . . . = { } and ( )- 2 n 1 2 | n Z n . Also, since E and O have no elements in common, = E S { }, the empty set . The empty set is the set which contains no elements....
View Full Document

Page1 / 9

sets_realnos_intervals - Sets, Real Numbers and Intervals...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online