2.1_SETS.pdf - Sets 1 What is a set • A set is a group of “objects” – – – – – People in a class Alice Bob Chris Classes offered by a

2.1_SETS.pdf - Sets 1 What is a set • A set is a group of...

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1 Sets
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2 What is a set? A set is a group of “objects” People in a class: { Alice, Bob, Chris } Classes offered by a department: { CS 101, CS 202, … } Colors of a rainbow: { red, orange, yellow, green, blue, purple } States of matter { solid, liquid, gas, plasma } Sets can contain non-related elements: { 3, a, red, Bob } Although a set can contain (almost) anything, we will most often use sets of numbers All positive numbers less than or equal to 5: {1, 2, 3, 4, 5} A few selected real numbers: { 2.1, π, 0, -6.32, e }
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4 The collection of persons living in Sorsogon is a set. Each person living in Sorsogon is an element of the set. The collection of all Brgy. in Bulan is a set. Each Barangay in Bulan is an element of the set.
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5 The roster method of specifying a set consists of surrounding the collection of elements with braces.
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6 For example the set of counting numbers from 1 to 5 would be written as {1, 2, 3, 4, 5}.
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7 A variation of the simple roster method uses the ellipsis ( … ) when the pattern is obvious and the set is large. {1, 3, 5, 7, … , 9007} is the set of odd counting numbers less than or equal to 9007. {1, 2, 3, … } is the set of all counting numbers.
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8 Set builder notation has the general form {variable | descriptive statement }. The vertical bar (in set builder notation) is always read as “such that”. Set builder notation is frequently used when the roster method is either inappropriate or inadequate.
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9 Example: {x | x < 6 and x is a counting number} is the set of all counting numbers less than 6. Note this is the same set as {1,2,3,4,5}. {x | x is a fraction whose numerator is 1 and whose denominator is a counting number }. Set builder notation will become much more concise and precise as more information is introduced.
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10 Set properties 1 Order does not matter We often write them in order because it is easier for humans to understand it that way {1, 2, 3, 4, 5} is equivalent to {3, 5, 2, 4, 1} Sets are notated with curly brackets
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11 Set properties 2 Sets do not have duplicate elements Consider the set of vowels in the alphabet. It makes no sense to list them as {a, a, a, e, i, o, o, o, o, o, u} What we really want is just {a, e, i, o, u} Consider the list of students in this class Again, it does not make sense to list somebody twice Note that a list is like a set, but order does matter and duplicate elements are allowed We won’t be studying lists much in this class
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