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3 5 2 ? 2 4 0 {(3,-2), (5,4), (-2,0)} x 3 5 -2 y -2 4 0 A relation is set of ordered pairs, where one number is mapped to another number. We need to be able to show relations in a variety of forms. In each form, there are pairs of numbers, where one number is considered the input and the other is the output. I. Mappings A Mapping looks like this: 3 5 2 2 4 0 This mapping contains three pairs: 3 is mapped onto -2 5 is mapped onto 4 -2 is mapped onto 0 II. Tables A Table looks like this: x 3 5 -2 y -2 4 0 This relation contains the same pairs as the mapping we just saw. (The names at the top of the table do NOT have to be x and y!) III. Coordinate Graphs A Coordinate Graph looks like this: (5, 4) (-2,0) (3,-2) To plot pairs on a coordinate plane, the first number tells you how far to the right or left of the origin to go, the second number is up or down. IV. Sets of Ordered Pairs A Set of Ordered Pairs looks like this: {(3,-2), (5,4), (-2,0)} We always put sets in braces { } and separate elements by commas. The order is easy to determine here! V. Input-Output Machines An Input-Output Machine looks like this: 3 ? -2 The arrows show which number is the first number (the input) and which is the second (the output). Re-Writing Relations This relation is in mapping form. Write a table for this relation. -2 3 -1 0 4 7 4 a -2 3 4 7 b 4 0 0 -1 NOTE: Some relations use the same values more than once. Re-Writing Relations This relation is in set form. Draw a mapping for this relation. {(1, 5), (0, 4), (4, 8), (-7, -3)} -7 0 -3 4 1 4 5 8 The last relation had a pattern to it. Did you notice? {(1, 5), (0, 4), (4, 8), (-7, -3)} Some relations are created by some set of operations, a formula that maps one number to another number. Formula Relations {(1, 5), (0, 4), (4, 8), (-7, -3)} In each pair, the second number is 4 more than the first number, the formula for this relation could be written: y = x + 4. Or use different letters: w = r + 4. A Formula Machine Input: 1 0 4 x+4 Output: 5 4 8 What formula creates this relation? 2 7 1 2 ? 4 14 1 What formula creates this relation? 1 2 7 2 2x 4 14 1 This relation follows a formula, find the formula & decide what 3 maps onto. -5 1 100 3 12 18 117 ? The formula is “add 17 to each input”, x + 17. 3 maps onto 20. -5 1 100 3 12 18 117 ? 20 What number maps onto 0? -5 1 100 -17 ? 12 18 117 0 Find the output values if the input values are {1, 3, 5, 7}. 1 x5 2 ?3 Input 1 Output 3 Find the output values if the input values are {1, 3, 5, 7}. 3 Input Output x5 2 ?4 1 3 3 4 Find the output values if the input values are {1, 3, 5, 7}. 5 Input Output x5 2 ?5 1 3 3 5 4 5 Find the output values if the input values are {1, 3, 5, 7}. 7 Input Output x5 2 6? 1 3 3 5 7 4 5 6