NOTES FOR SET THEORY UNIT

SETS: HISTORY

Georg 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.

SETS: VOCABULARY

set- a collection of objects

elements, or members - things in a set

subsets- a part of another set

proper 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

empty 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

Sets of Numbers: VOCABULARY

natural 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. Includes whole numbers, natural numbers, integers, fractions , decimals that repeat or terminate.

irrational numbers- numbers that can not be written as a fraction. , 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 2

prime- a number with exactly 2 factors, 1 and itselfIs one a prime? No.

composite- a number with more than 2 unique factors.

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

NOTATION

3 Ways to Describe Sets

Words / List Elements / Set Builder Notation
The set of all teachers in the room. / {McAllister} / { is a certified teacher}
The set of even numbers between 1 and 9. / {2, 4, 6, 8} / { is an even number between 1 and 9}
{ and x < 9 and even}

Examples:

A = {a, b, c, d, e, f, g, h}= UB = {a, b, c}C = {a, c, e, g}

D = {c, d, e, f}E = {b, d, f, h}

reads reads reads

'B is a subset of A''B is a proper subset of A'‘a is an element of A”

The empty set is written: { } =

Special Cases

- every set is a subset of itself- every set has the empty set as a subset

SET OPERATIONS: VOCABULARY

intersection- element of 2 sets that the sets have in common,

Example: A = {a, b, c, d, e, f, g, h}= UB = {a, b, c}C = {a, c, e, g}

D = {c, d, e, f}E = {b, d, f, h}

B C = {a, c}What is B E?

union- put all elements of 2 sets together into a new set,

Example: B C = {a, b, c, e, g}

complement- the elements of the universal set not in a given set,

Example: = {b, d, f, h}

= E

disjoint sets- have no common elements (intersection is empty)

difference of sets – the set difference of set B from set A, written A – B is the set of all elements in A that are not in B. Example: A – B = {d,e,f,g,h}

ANOTHER EXAMPLE

Given: U = {1, 2, 3, 4, 5, 6} A = {2, 4, 6}B = {1, 3, 5}C = {2, 3, 4}

True or False:

A Cfalse

true

false

true

true

A – C = { 3, 6}false

SET REPRESENTATIONS

Venn diagrams are used to represent the relationships between sets in a graphical manner. Venn diagrams may show the exact elements of sets, the general relationship between sets, or the number of elements in each part of the relationship between sets.

EXAMPLES USING VENN DIAGRAMS- draw a Venn diagram to represent these sets

1. U = {1, 2, 3, …, 10}, A = , B = ,

C =

2. Draw a Venn diagram of the sets describe in the section on Set Operations

Venn diagrams to solve cardinality problems.

1. Twenty-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?

2. In 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?

3. 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?