SETS



WRITING SETS

Ways to Write Sets:

o Roster or a list of the actual elements (within brackets)

Rule: set described in words (within brackets)

Set Builder Notation similar to rule but more formal

Interval Notation used if a set is continuous and could be also represented as an inequality

Graphing a line (inequality) is a visual way to represent a continuous number set

Venn diagram used with multiple set where parts of the sets overlap

|Interval Notation:  (description) |(graphic) |

|Open Interval:   (1, 5)  |(1, 5) |

|is the inequality 1 < x < 5  |[pic] |

|where the endpoints are NOT included. | |

|Closed Interval:  [1, 5]  |[1, 5] |

|is the inequality 1 < x 1 |[pic] |

|where 1 is not included | |

|infinity is always expressed as being "open" (not included). | |

|Non-ending Interval:  (-(, 5] | |

|is the inequality x < 5 | |

|where 5 is included | |

|infinity is always expressed as being "open" (not included). | |

|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 |Inequality: 1 < x < 5 |

|five." | |

|"x is less than or equal to 5 and greater than or equal to 1" |Set Builder: { x [pic][pic]| 1 < x < 5} |

|"x is between 1 and 5, inclusive" |Interval: [1,5] |

-----------------------

If there is an extensive number of elements in a set, the rule form is more practical than the roster form.

Writing a very large roster list would be too time-consuming.

Rule Form

{one-digit prime numbers}

{the letters of the word Mississippi}

{states of the U.S. that touch the Pacific Ocean}

Roster Form

{2, 3, 5, 7}

{i, m, p, s}

Note: The letters are not repeated.

{California, Oregon, Washington, Alaska, Hawaii}

Set Builder Notation: A way of describing a set in “mathematical shorthand” without listing the elements of the set. Set builder notation is similar to rule form, but it is considered to be a more precise and “formal” way to describe a set.

Example (1): Describing the set of all the natural (counting numbers).

Set Builder Notation: {x( x ( [pic]} or {x : x ( [pic]}

Translation: “all x , such that, x is an element of the natural numbers”

Example (2): Describing the set of multiples of 5.

Set Builder Notation: {m (m is a multiple of 5} or

{m : m is a multiple of 5}

Translation: “all m, such that, m is a multiple of 5”

Interval Notation: An interval is a connected subset of numbers. Inequalities are examples of interval subsets.

Interval notation is an alternate way to write an inequality instead of using the symbols (, (, (, or ( , or graphing it on the number line.

Symbols used in interval notation:

( or ) means “not included in the set”

[ or ] means “included in the set”

-[pic] means “negative infinity”

[pic] means “positive infinity”

Example (1): The inequality 2 ( x ( 6

Interval Notation: [ 2, 6 )

Translation: “all real numbers in the interval of 2 to 6, including 2 and excluding 6”

Example (2): The inequality x ( 5

Interval Notation: ( 5, [pic])

Translation: “all real numbers greater than 5”

Note: Use the “not included” symbol when dealing with infinity and negative infinity since you can’t ever reach the end of either.

You may see a set written in any of the formats we have discussed.

The following is an example of the same exact set written:

▪ In words

▪ As an inequality

▪ In set builder notation

▪ In interval notation

▪

................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download