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Transcript
1 2 Sets and Functions Introduction to Set Theory In Mathematics, the word set refers to a group of numbers or other types of elements. Sets are written as follows: { 1, 2, 3, 4, 5, 6 } The symbol { -0.7, -0.2, 0.1 } means ‘is an element of’. Examples 4 { 1, 2, 3, 4, 5 } 7 { 1, 2, 3 } { 6, 7, 8 } ⊆ { 6, 7, 8, 9 } { red, green, blue } The Basic Number Sets N Natural numbers { 1, 2, 3, 4, 5, ... } W Whole numbers { 0, 1, 2, 3, 4, 5, ... } Z Integers { ... -3, -2, -1, 0, 1, 2, 3, ... } Q Rational numbers Includes all integers, plus any number which can be written as a fraction. R Real numbers Includes all rational numbers, plus irrational numbers such as √ 7 or π . Set Theory and Venn Diagrams Venn Diagrams are illustrations which use overlapping circles to display logical connections between sets. R 5 7 Z N W Q √ 82 Functions and Mappings What is a function? A function, f, consists of : 1. a formula, f(x), which tells you what to do with a given value of x. 2. a domain which describes the values of x you are allowed to use in the formula. Important Each number in the domain has one output number in the range. INPUT domain f (x) OUTPUT range The ‘input’ is also known as the domain of the function, with the ‘output’ referred to as the range. Functions and Arrow Diagrams A •1 •2 •3 B •3 •7 •9 Domain Range This is known as an arrow diagram. Diagram 1 is a function as each element in set A is mapped to one and only one element in set B. A •1 •2 •3 B •3 •7 •9 This is not a function as 2 is mapped to 7 and 9. Also 3 does not appear to be mapped to anything. Function Graphs A typical graph of a function f shows the points (a, f(a)) for all values x = a in the domain of f. y = f(x) range (‘out’ numbers) f(a) (a, f(a)) a domain (‘in’ numbers) The Vertical Line Test How can I decide from a graph whether it is a function? The vertical line test is a way to determine whether or not we have a function. If a vertical line intersects the graph in more than one place, then it is NOT a function. The test is simply a restatement of the definition of a function which states that every x value must have a unique y value. HHM Exercise 2B Page 24 Q’s 1 and 8 Success Criteria :use set notation Illustrate vertical line test