Set Theory - Half Hollow Hills Central School District

[Pages:19]Set Theory:

Topics of Study:

Set Notation: Roster notation, set-builder notation, interval Notation

Set Notation: Operations with sets Union, Intersection, and Complement

Videos to introduce set theory including operations of sets:

Tutorials 1. Video for interval notation/set builder notation/roster notation

2. Video for interval notation

3. Introduction to sets: Intersection and Union

4. Video for all operations of sets

Extra practice/interactive links: 1. Types of sets and set notation practice 2. Union and Intersection of sets 3. Cumulative Review 4. Union and Intersection Extra Practice with Answer Key

Introduction to Set Theory: Set Notation

Set Theory: A set is a collection of unique elements. Elements in a set do not "repeat".

Methods of Describing Sets: Sets may be described in many ways: by roster, by set-builder notation, and by interval notation

Method 1: Roster Notation: A roster is a list of the elements in a set, separated by commas and surrounded by French curly braces.

is a roster for the set of integers from 2 to 6, inclusive (including all numbers) is a roster for the set of positive integers. The three dots indicate that the numbers continue in the same pattern indefinitely. The set continues towards positive infinity

Method 2: Set-builder notation: Set-builder notation is a mathematical shorthand for precisely stating all numbers of a specific set that possess a specific property. (using inequalities)

= real numbers; = integer numbers; = natural numbers.

is set-builder notation for the set of integers from 2 to 6,

inclusive.

= "is an element of"

Z = the set of integers

| = the words "such that"

The statement is read, "all x that are elements of the set of integers, such

that, x is between 2 and 6 inclusive."

The statement is read, "all x that are elements of the set of integers, such

that, the x values are greater than 0, positive." (The positive integers can also be indicated as the set Z+.)

It is also possible to use a colon ( : ), instead of the | , to represent the words "such that".

is the same as

Method 3: By interval notation: An interval is a connected subset of numbers. Interval notation is an alternative to expressing your answer as an inequality. Unless specified otherwise, we will be working with real numbers.

When using interval notation, the symbol: ( means "not included" or "open". [ means "included" or "closed".

as an inequality. in interval notation.

Interval Notation: (description) Open Interval: (a, b) is interpreted as a < x < b where the endpoints are NOT included. (While this notation resembles an ordered pair, in this context it refers to the interval upon which you are working.)

Closed Interval: [a, b] is interpreted as a < x < b where the endpoints are included.

Half-Open Interval: (a, b] is interpreted as a < x < b where a is not included, but b is included.

Half-Open Interval: [a, b) is interpreted as a < x < b where a is included, but b is not included.

(diagram) (1, 5)

[1, 5] (1, 5] [1, 5)

Non-ending Interval:

is interpreted as x > a where a is not

included and infinity is always expressed as being "open" (not

included).

Non-ending Interval:

is interpreted as x < b where b is

included and again, infinity is always expressed as being "open"

(not included).

As an inequality: In interval notation:

x < 0 or 2 < x < 10

In Summary:

The following statements and symbols ALL represent the same interval:

WORDS:

SYMBOLS:

"all numbers between positive one and positive five, including the one and the five."

1 < x < 5

"x is less than or equal to 5 and greater than or equal to 1"

{ x | 1 < x < 5}

"x is between 1 and 5, inclusive"

[1,5]

Practice: Putting it all together:

x x 3orx 5

Operations of Sets

Remember: Sets are simply collections of items. A set may contain your favorite even numbers, the days of the week, or the names of your brothers and sisters. The items contained within a set are called elements, and

elements in a set do not "repeat".

The elements of a set are often listed by roster notation. A roster is a list of the elements in a set, separated by commas and surrounded by French curly braces.

**The empty set is denoted with the symbol: Examples:

Let set A be the numbers 3, 6, 9. A = {3, 6, 9} (in roster notation)

(5 is not in set A)

Let B = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} A = {3, 6, 9}

Set A is a subset of set B, since every element in set A is also an element of set B. The notation is:

Operations:

1. Intersection of sets: The Venn diagram below shows two sets A and B that overlap. The universal set is U. Values that

belong to both set A and set B are located in the center region labeled overlap. This region is called the "intersection" of the two sets. (Intersection, is only where the two sets intersect, or overlap.)

where the circles

2. Union of Sets: The notation

represents the entire region covered by both sets A and B (and the

section where they overlap). This region is called the "union" of the two sets.

(Union, like marriage, brings all of both sets together.)

3. Complement of Sets: If we cut out sets A and B from the picture above, the remaining region in U, the

universal set, is labeled ,

and is called the complement of the union of sets A and B.

A complement of a set is all of the elements (in the universe) that are NOT in the set.

NOTE*: The complement of a set can be represented with several differing notations.

The complement of set A can be written as

Example: Let U (the universal set) = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} (a subset of the positive integers) A = {2, 4, 6, 8} B = {1, 2, 3, 4, 5}

Union - ALL elements in BOTH sets

Intersection - elements where sets overlap

Complement - elements NOT in the set

Extra Practice of Complements:

Extra Practice with Union and Intersection:

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