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Set Theory. Jim Williams HONP-112 Week 3. Set Theory. Set Theory is a practical implementation of Boolean logic that examines the relationships between groups of objects .

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Set Theory

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Set Theory

Jim Williams

HONP-112

Week 3


Set Theory

  • Set Theory is a practical implementation of Boolean logic that examines the relationships between groups of objects.

  • Set theory has numerous real-life applications in computer systems design, as well as database searching (we will learn more about databases later, but will touch on search concepts today)


Definitions and Conventions

  • A set consists of individual elements.

  • A set is denoted by curly brackets and elements are separated by commas:

    • {A,E,I,O,U}

  • A set that has no elements is called the empty set, AKA the null set.

    • {}


Universal Set

  • The Universal Set (or “Universe”) contains all possible values of whatever type of objects we are studying.

  • Sets can be infinite (i.e. {all real numbers}), or finite (i.e. {all letters of the alphabet}).

  • We will only be studying finite (AKA "discrete") sets.

  • We will use Uto denote the universal set. Do not confuse this with the UNION symbol (later).


Subsets

  • Lets assume we have 2 sets A and B.

  • B is a subset of A if and only if all the elements in B are also in A.

  • Every set is a subset of the Universal Set.

  • Example: {1,2,6} is a subset of {1,2,3,4,5,6}


Venn Diagram

UNIVERSE

SET

A

SET

B

A Venn Diagram can be used to graphically illustrate the relationship between sets.


Union

UNIVERSE

SET

A

SET

A

SET

B

SET

B

SET A UNIONSET B contains all the elements that are either in A or B. The shaded areas illustrate the union.


Intersection

UNIVERSE

SET

A

SET

B

SET A INTERSECTSET B contains only the elements that are in A and also in B. The shaded area illustrates the intersection.


Compliment

UNIVERSE

SET

A

SET

B

SET A COMPLIMENT contains only the elements that are not in a given set. The shaded areas illustrate the compliment of A.


Symbols for Set Operators

  • Union: ⋃

    • Example: A ⋃ B

  • Intersection: ⋂

    • Example: A ⋂ B

  • Compliment: '

    • Example: A'


Set Example for next 3 slides

  • Define Universal Set = All U.S. Coins

  • U={penny, nickel, dime, quarter, half-dollar, dollar}

  • Set A = {penny, nickel}

  • Set B = {nickel, dime, dollar}


Set Union - Example

  • U={penny, nickel, dime, quarter, half-dollar, dollar}

  • Set A = {penny, nickel}

  • Set B = {nickel, dime, dollar}

  • A ⋃ B = {penny, nickel, dime, dollar}

  • IMPORTANT: Notice that the set elements never repeat within a single set of any kind (see there is only one “nickel” element in the result set!) Do not forget this!


Set Intersection - Example

  • U={penny, nickel, dime, quarter, half-dollar, dollar}

  • Set A = {penny, nickel}

  • Set B = {nickel, dime, dollar}

  • A ⋂ B = {nickel}


Set Compliment - Example

  • U={penny, nickel, dime, quarter, half-dollar, dollar}

  • Set A = {penny, nickel}

  • Set B = {nickel, dime, dollar}

  • A' = {dime, quarter, half-dollar, dollar}


Work out some examples

  • U={penny, nickel, dime, quarter, half-dollar, dollar}

  • Set A = {all denominations over 10 cents.}

  • Set B = {all coins not silver in color}

  • What is A ⋃ B? A ⋂ B?

  • Try some more examples.


Set Theory Vis-à-vis Boolean logic

  • The UNION set operator functions in a similar manner to the Boolean OR logical operator.

  • The INTERSECTION set operator functions in a similar manner to the Boolean AND logical operator.

  • The COMPLIMENT set operator functions in a similar manner to the Boolean NOT logical operator.


The UNION operator as OR

  • Given Set A , Set B

  • The UNION applies a boolean OR to each element of each set.

  • A result of True (1) for any of these cases qualifies the element to be included in the result set.

  • Let’s Look at an example…


The UNION operator as OR

  • Set A = {penny, nickel}

  • Set B = {nickel, dime, dollar}

  • The OR is applied to each element in set A and the corresponding element in set B

  • Results with 1 are included in the result set.

  • A ⋃ B = {penny, nickel, dime, dollar}


The INTERSECT operator as AND

  • Given Set A , Set B

  • The INTERSECT applies a boolean AND to each element of each set.

  • A result of True (1) for any of these cases qualifies the element to be included in the result set.

  • Let’s Look at an example…


The INTERSECT operator as AND

  • Set A = {penny, nickel}

  • Set B = {nickel, dime, dollar}

  • The AND is applied to each element in set A and the corresponding element in set B

  • Results with 1 are included in the result set.

  • A ⋂ B = {nickel}


The COMPLIMENT as NOT

  • Given Set A

  • The COMPLIMENT applies a boolean NOT to each element of the set.

  • But, remember, that we also have to consider the entire universal set.

  • This is because a NOT is a unary operator. In this context, it only operates on the elements of a single set. But there are elements in the universal set that are still NOT in A.

  • (This type of compliment is the “absolute compliment” - which is the only type we are concerned with here).

  • Let’s look at an example…


The COMPLIMENT operator as NOT

  • Set A = {penny, nickel}

  • The NOT is applied to each element in the set we are taking the compliment of. Remember we are still doing this in relation to the universal set!

  • Results with 1 are included in the result set.

  • A' = {dime, quarter, half-dollar, dollar}


Alternate way of thinking about the Set Compliment (advanced)…

  • The COMPLIMENT applies a NAND against each element of the universal set and the corresponding elements of the set we are taking the compliment of.

  • Set A = {penny, nickel}


Set theory and bit-wise mathematics

  • The concept we have illustrated in our previous tables is also used in various other computer circuits, and is called BIT-MASKING.

  • Given a sequence of bits, and a mask (also made up of some chosen sequence of bits), we can apply an AND mask, an OR mask, etc.

  • We need not worry about what bit-masking is used for right now – or why certain bit sequences may be chosen as a mask.

  • But we should know HOW to apply a mask to a bit sequence.


The AND mask

  • The resulting bit sequence results from applying an AND against each bit in the sequence, and the corresponding bit in the mask. Result is 00110000.

  • Given a BIT sequence 00110110, and a MASK of 11110000:


The OR mask

  • The resulting bit sequence results from applying an OR against each bit in the sequence, and the corresponding bit in the mask. Result is 11110110.

  • Given a BIT sequence 00110110, and a MASK of 11110000:


Bitwise Masks: Observation

  • Given a BIT sequence 00110110, and a MASK of 11110000:

  • The AND mask results in 00110000

  • The OR mask results in 11110110

  • So, which type of MASK results in having more 1s in the resulting sequence?


Set Theory applied to Databases

  • We can better understand the relationship between Boolean Logic and Set Theory by using a database example.

  • We will study databases in more depth later – but just keep the concept in mind.

  • Boolean searches are done against large databases in many situations (think of some).


Search Operators

  • Databases usually use search operators that correspond with the standard Boolean operators of AND, OR, and NOT.

  • BUT – in the case of searches, they are applied to whether an item being searched for meets certain criteria.


Search Operators and Set Theory

  • To understand this better, we need to think of search criteria in a different way.

  • Keep in mind that each search criterion will really be creating a separate SUBSET that meets the criterion.

  • When we search, we can apply boolean operators to connect criteria together in different ways.

  • So what we are really doing is creating different subsets and applying set operators to them.


The OR (Union) Database Search Operator: Example

  • The OR search operator combines 2 or more subsetsinto a single largersubset.

  • Example: Criterion 1: Customers from the 07463 zipcode. Criterion 2: Customers who have made a purchase within the past month.

  • (Criterion 1) OR (Criterion 2) will create a single subset containing all customers from 07463, along with all the customers who have made a purchase within the past month, regardless of their zip code.


The AND (Intersection) Database Search Operator: Example

  • The AND search operator selects the records that 2 or more subsets have in common into a single smaller subset.

  • Example: Criterion 1: Customers from the 07463 zipcode. Criterion 2: Customers who have made a purchase within the past month.

  • (Criterion 1) AND (Criterion 2) will create a single subset containing the customers from 07463, who also have made a purchase within the past month.


The NOT (Compliment) Database Search Operator: Example

  • The NOT search operator gives us all the records in a set that do not belong to a particular subset.

  • Example: Criteria 1: Customers from the 07463 zip code.

  • NOT (Criteria 1) will create a single subset containing the customers who are NOT from 07463.


More practical examples: Library Searching

  • In your college studies you will frequently need to look up books and articles in the library catalog, based on certain criteria (conditions).

  • The following slides will illustrate some examples of this and hopefully clarify the relationship between boolean search operators and their corresponding “behind the scenes” set operations.

  • Remember that in these examples the Universal Set is the entire library collection.


Library search example - AND

  • Consider this: we want to find all books written by Stephen King within the past 5 years.

  • What we are really doing: Set A: Books written by Stephen King. Set B: All books written within the past 5 years.

  • If we want books that meet BOTH criteria, what we are looking for is the INTERSECTION between sets A and B.

  • A library system may allow us to say something like: Author=Stephen King AND date >= 2007.


Library search example - OR

  • But what if we wanted any book written by Stephen King, or any book (regardless of author) written within the past 5 years.

  • What we are really doing: Set A: Books written by Stephen King. Set B: All books written within the past 5 years.

  • If we want books that meet EITHER criteria, what we am looking for is the UNION of sets A and B.

  • A library system may allow us to say something like: Author=Stephen King OR date >= 2007.


Library search example - NOT

  • But what if we didn’t care for Stephen King’s writing? So, we wanted to find any book NOT written by him.

  • What we are really doing: Set A: Books written by Stephen King.

  • If we want books that are NOT in set A, what we are looking for is the COMPLIMENT of set A.

  • A library system may allow us to say something like: Author IS NOT Stephen King


Library search example - complex

  • In real life most library searches are more complex.

  • Example: Mystery books written in the past 5 years, but not by Stephen King.

  • Set A: All Mystery Books. Set B: All Books written during the past 5 years. Set C: All Books written by Stephen King.

  • A library system might let us do something like this: Category=Mystery AND Date >= 2007 AND Author IS NOT Stephen King.

  • In set theory terms, this means: (A ⋂ B) ⋂ C'


Review

  • Know the three set operators

  • Know how these are related to the three basic boolean operators.

  • Use these concepts to solve set theory problems.

  • Use these concepts to apply a given mask to a given bit sequence.

  • Understand how these concepts are applied to boolean searching of databases.


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