What is discussed in this lecture?

1.   Interval Notation
2.   Set-Builder Notation

Both the Interval and Set-Builder Notations are used to efficiently express a certain span of numbers on the number line. 

1.   Interval Notation

In Interval Notation, we use parentheses, brackets, and sometimes the "union" symbol. 

Brackets [ ] :  In Interval Notation, a bracket next to a number indicates that the number is included in the solution interval.

For example, would be a representation of the interval on the number line between and inclusive. 

NOTE: When using Interval Notation, the smaller number is ALWAYS written first, followed by the larger number.  would NOT be a correct representation of the interval.

Parentheses ( ) :  In Interval Notation, a parenthesis next to a number indicates that the number is NOT included in the solution interval.  Negative and positive infinity always start or end, respectively, with a parenthesis. 

For example, would be a representation of the interval on the number line between and , but NOT including and .

For example, or or would be a representation of intervals including the infinity symbol.

Union :  In Interval Notation, the symbol joins two intervals of numbers.

For example, would be a representation of two joined intervals on the number line. The first interval represents the numbers between and , but not including the numbers. The second interval represents the numbers between and , but not include the numbers. 

2.   Set-Builder Notation

In Set-Builder Notation we use mathematical symbols, braces, and a vertical bar separator. 

Mathematical Symbols:  >, <, , , ,

>  means "greater than"
means "greater than or equal to"
<  means "less than"
means "less than or equal to"
means "not equal to"

Braces { } :  In Set-Builder Notation, a brace starts and ends the set.

Vertical bar | :  In Set-Builder Notation, the vertical bar is translated into English as "such that".

Example 1: 

Write in Interval and Set-Builder Notation.

Interval Notation:

It is an open interval along the x-axis starting at 2, but not including 2 and going to 6, but not including 6.
Note that the smaller number is always written first!

Set-Builder Notation:

Read the part in braces as "the set of all x such that x is greater than 2 but less than 6."

Example 2: 

Write in Interval and Set-Builder Notation.

Interval Notation:

It is a closed interval along the x-axis starting at 2 inclusive and going to 6 inclusive.

Set-Builder Notation:

Read the part in braces as "the set of all x such that x is greater than or equal to 2 but less than or equal to 6."

Example 3: 

Write in Interval and Set-Builder Notation.

Interval Notation:

It is a half-open interval along the x-axis starting at 2 inclusive and going to 6, but not including 6.

Set-Builder Notation:

Read the part in braces as "the set of all x such that x is greater than or equal to 2 but less than 6."

Example 4: 

Write in Interval and Set-Builder Notation.

Interval Notation:

It is a interval along the x-axis starting at 2, but not including 2, and extending to positive infinity.
Negative and positive infinity always start or end, respectively, with a parenthesis.
Note that the smaller number is always written first!  In this case 2 is smaller than positive infinity.

Set-Builder Notation:

Read the part in braces as "the set of all x such that x is greater than 2".

Example 5: 

Write in Interval and Set-Builder Notation.

Interval Notation:

It is an interval along the x-axis starting at negative infinity and extending to 2, but not including 2.
Negative and positive infinity always start or end, respectively, with a parenthesis.
Note that the smaller number is always written first!  In this case, negative infinity is smaller than 2.

Set-Builder Notation:

Read the part in braces as "the set of all numbers x such that x is less than or equal to 2".

Example 6: 

Write All Real Numbers in Interval and Set-Builder Notation.

Interval Notation:

It is an interval along the x-axis starting at negative infinity and extending to positive infinity

Set-Builder Notation:

Read the part in braces as "the set of all x such that x consists of all rational and irrational numbers.

Example 7: 

Write All Real Numbers except 1 and 5 in Interval and Set-Builder Notation.

Interval Notation:

Here we had to write three separate intervals and them join them with a union symbol in order to leave out the numbers 1 and 5.

Set-Builder Notation:

When we leave out a few finite number, it is easier to write th interval in Set-Builder Notation. Read the part in braces as "the set of all x such that x is not equal to 1 and not equal to 5."

Example 8: 

Write but in Interval and Set-Builder Notation.

Interval Notation:

Here we had to write two separate intervals and then join them with a union symbol in order to leave out the number 0.

Set-Builder Notation:

Read the part in braces as "the set of all x such that x is less than 2 and not equal to 0."