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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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