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# Intro To Abstract Math MA 315

Purdue University Calumet

GPA 3.93

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This 0 page Class Notes was uploaded by Vergie Ankunding on Sunday November 1, 2015. The Class Notes belongs to MA 315 at Purdue University Calumet taught by Peter Turbek in Fall. Since its upload, it has received 10 views. For similar materials see /class/232684/ma-315-purdue-university-calumet in Mathematics (M) at Purdue University Calumet.

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Date Created: 11/01/15

MA 315 Spring 2009 Set Theory Basics Sets are often denoted by roman capitals such as A B C and so on The fundamental thing to say about a set A is whether something else z is an element of A If m is an element of A we write m E A This is a statement Sets are completely determined by their elements A B means that A and B have the same elements More formally we can write this as xeA xEB We say that A is a subset of B if all the elements of A are also elements of B so zEA zEBWewritethisACB Notation If we have a list of all the elements of a set we enclose the list in braces to denote the set itself for example 158 1320 If we have a description of the elements of a set we enclose the description in braces to denote the set itself even integers If a set consists of the elements of another set satisfying a certain property we use set builder notation inside braces we put a colon or vertical slash To the left of this is a variable element of the larger set and to the right is the property the variable must satisfy 71 E Z 16 divides n2 71 If we think of the elements of a set as being those of a certain form in another set we put the form on the left of the colon or vertical slash and the variable on the right 4k3k Z Exercise Prove or disprove 4k3k Z4k71k Z MA 315 Spring 2008 Handout 1 Here are some useful de nitions Integers The set of integers is denoted by Z and is the set 73 72 71 0 1 23 Rational Numbers The set of rational numbers is denoted by Q and is the following set glaEZbEZby O Real Numbers The real numbers are denoted by R Complex Numbers The complex numbers are denoted by C and is the following set abila E Rb E R Note that all elements of Z are elements of Q all elements of Q are elements of R and all elements of R are elements of C De nition Let a and b be real or complex numbers and assume b 31 0 We de ne a divided by b denoted by ab to be the number 0 such that be 1 De nition Let a and b be integers and assume b 31 0 We say b divides a and write bla if there exists an integer c such that be a If bla we also say that a is divisible by b Below is a proof for you to critique Theorem If n is even then 712 is even Proof Let n be any integer so 271 represents any even integer Then 712 2702 2n Since 271 is an even number and by de nition the result of two even numbers multiplied together is also even we have proved that if n is even then 712 is even

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