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Transcript
WRITING SETS Ways to Write Sets: 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 Roster Form {2, 3, 5, 7} {i, m, p, s} Note: The letters are not repeated. {California, Oregon, Washington, Alaska, Hawaii} 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 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} Set Builder Notation: A way of describing a set in Interval Notation: An interval is a connected subset of “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. 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. Example (1): Describing the set of all the natural (counting numbers). Set Builder Notation: {x x Ν } or {x : x Ν } 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” Note: Use the “not included” symbol when dealing with infinity and negative infinity since you can’t ever reach the end of either. Symbols used in interval notation: ( or ) means “not included in the set” [ or ] means “included in the set” - ∞ means “negative infinity” ∞ 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, ∞) Translation: “all real numbers greater than 5” Interval Notation: (description) (graphic) Open Interval: (1, 5) is the inequality 1 < x < 5 where the endpoints are NOT included. (1, 5) Closed Interval: [1, 5] is the inequality 1 < x <5 where the endpoints are included. [1, 5] Half-Open Interval: (1, 5] is the inequality 1 < x < 5 where 1 is not included, but 5 is included. (1, 5] Half-Open Interval: [1, 5) is the inequality 1 < x < 5 where 1 is included, but 5 is not included. [1, 5) Non-ending Interval: (1,) is the inequality x > 1 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). 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 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." Inequality: 1 < x < 5 "x is less than or equal to 5 and greater than or equal to 1" Set Builder: { x "x is between 1 and 5, inclusive" Interval: [1,5] | 1 < x < 5}