Transcript 1.1 & 1.2
1.1 – SETS AND SYMBOLS Goals SWBAT understand basic set notation and set symbols SWBAT solve simple sentences with a given domain SWBAT graph sets of numbers on a number line. IMPORTANT DEFINITIONS A set is a well-defined collection of objects. Each object in the set is called an element. Set notation uses braces. Example: 1, 3, 8,13 WAYS TO SPECIFY A SET 1. List the names of its members within braces. (roster) Example: 1, 2, 3 2. A rule or description of its members. Example: all numbers greater than 1 3. By a graph. (Covered in 1.2) COMMON SET SYMBOLS Read as: “the set whose members are” Meaning: a collection or set Read as: “is a member of” or “is an element of” Meaning: is in the collection or set COMMON SET SYMBOLS A B Read as: “A is a subset of B” Meaning: Every member of A is a member of B. For every set A, A is a subset of A. or Read as: “the null set” or “the empty set” The set that contains no elements. The empty set is considered a subset of every set. COMMON SET SYMBOLS AB Read as: “A is equal to B” Meaning: A and B contain exactly the same elements / Read as: “is not” Meaning: used with other symbols to show negation: , , COMMON SET SYMBOLS AB Read as: “A intersect B” Meaning: The set of elements belonging to both A and B. AB Read as: “A union B” Meaning: The set of elements in at least one of the given sets. TRUE OR FALSE? 1 3 1 2, , , 4 2 3 TRUE OR FALSE? 3, 4,5 odd integers TRUE OR FALSE? 1 1 1 0.25 , , 0, 2 4 3 TRUE OR FALSE? real numbers integers 1.2 – OPEN SENTENCES AND GRAPHS DEFINITIONS An expression is a number, a variable, or a sum, difference, product, or quotient that contains one or more variables. Examples: 7 9, 4 x , 3 5 y 7 2, 4 12 DEFINITIONS A variable is a symbol, usually a letter, that represents any of the members of a specified set. This set is called the domain of the variable, and its members are called the values of the variable. DEFINITIONS A mathematical sentence is a group of symbols that states a relationship between two mathematical expressions. These can be either true or false. Examples: 3 2 , 1 2 17, 14 8 6 4 5 DEFINITIONS An open sentence is a mathematical sentence that contains one or more variables. An open sentence cannot be determined true or false without knowing what value the variable represents. Examples: x 7 4, m p 2m 5, h 93 DEFINITIONS The values of the variable that make an open sentence true are called the solutions of the open sentence. The solution set is the set of all solutions that make the open sentence true. To solve an open sentence over a given domain, find the solution set using this domain. QUESTIONS 1-4: SOLVE THE OPEN SENTENCE 2,3,4,5 OVER THE DOMAIN 1. 2x 3 7 2. 8t 6t 2t 3. n 1 is an integer 3 4. y y 1 DEFINITIONS A real number is any number that is positive, negative, or zero. Subsets of Real Numbers: Natural Numbers:1,2,3,... Whole Numbers: 0,1,2,3,... Integers: ...,3,2,1,0,1,2,3,... It can be useful to graph the solution set of an open sentence on a number line. 5. Graph each subset of the real numbers on a number line. a. Natural Numbers b. Whole Numbers c. Integers GRAPH EACH SET OF NUMBERS ON THE NUMBER LINE. 6. 3 5 ,1, ,3 2 2 7. The set of integers that are multiples of 4. SOLVE EACH OPEN SENTENCE OVER THE SET OF POSITIVE INTEGERS AND GRAPH THE SOLUTION SET. 8. 25 y 9 9. z 5 10. 2 2 h 2 Is an integer