Set theory for teachersSet theory for teachersMA118Summer 2008McAllister

Background of set theoryBackground of set theoryGeorg Cantor (1845-1918).Big component of ‘new math’

curriculums that were pervasive in 60’s and early 70’s.

The most fundamental of concepts upon which all of mathematics is grounded

Important for teachers to understand the language and basics of sets.◦Understanding teachers’ manuals and

research◦Add dept to your understanding of math

VocabularyVocabularyset- a collection of objectselements, or members - things in a set

subsets- a part of another setproper subsets- a subset that is not equal to the set it is a part of

Universal set – The set of all possible elements under consideration

More termsMore termsempty set(null)- set with no members

well-defined sets-sets that are described clearly so there's no question of membership

Finite sets – sets that can be counted

Infinite set – sets that cannot be counted

Examples of number setsExamples of number setsnatural numbers- {1, 2, 3, 4, …} (counting numbers)

whole numbers- {0, 1, 2, 3, 4, …}

integers – The set of whole numbers and their opposites.

rational numbers- numbers that can be written as a fraction.

More examples of number More examples of number setssets irrational numbers- numbers that can not be

written as a fraction, OR non-repeating decimals, non-terminating decimals

real numbers- numbers that can describe a distance, numbers that form a one-to-one correspondence with the points on a number line. Composed of the rationals and irrationals.

evens-divisible by 2odds-not evenly divisible by 2prime- a number with exactly 2 factors, 1 and

itself Is one a prime? No.composite- a number with more than 2 unique

factors.multiples- exp: multiples of 2={2, 4, 6…}

Notation – very importantNotation – very importantWords

◦The set of all teachers in the room.◦The set of even numbers between 1

and 9List Elements

◦A = {McAllister}◦B = {2, 4, 6, 8}

Set Builder Notation◦A = { x1 x is a certified teacher} ◦B = {x1x is even and 1 < x < 9 }

Other notation and Other notation and definitionsdefinitionsIt’s time to move to the

document camera for this part.

Venn diagram problemsVenn diagram problemsTwenty-four dogs are in a kennel.  Twelve

of the dogs are black, six of the dogs have short tails, and fifteen of the dogs have long hair.  There is only one dog that is black with a short tail and long hair.  Two of the dogs are black with short tails and do not have long hair.  Two of the dogs have short tails and long hair but are not black.  If all of the dogs in the kennel have at least one of the mentioned characteristics, how many dogs are black with long hair but do not have short tails?

Disjoint set problemDisjoint set problemIn a group of 37 people, 18 are neither women nor lawyers. Ten are women and 13 are lawyers. How many lawyers in the group are not women?

One moreOne more

In a survey of 6500 people, 5100 had a car, 2280 had a pet, 5420 had a T.V., 4800 had a T.V. and a car, 1500 had a T.V. and a pet, 1250 had a car and a pet, and 1100 had a T.V., car, and pet. How many people had no car, no T.V. and no pet?