Set Properties - Discrete Mathematics - Lecture Slides, Slides of Discrete Mathematics

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Sets

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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 }
  • States in the US: { Alabama, Alaska, Virginia, … }
  • Sets can contain non-related elements: { 3, a, red, Virginia }
• 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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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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Specifying a set 1

• Sets are usually represented by a capital letter

(A, B, S, etc.)

• Elements are usually represented by an italic

lower-case letter ( a , x , y , etc.)

• Easiest way to specify a set is to list all the

elements: A = {1, 2, 3, 4, 5}

• Not always feasible for large or infinite sets

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Specifying a set 3

• A set is said to “contain” the various

“members” or “elements” that make up the

set

• If an element a is a member of (or an element of) a set S, we use then notation a ∈ S - 4 ∈ {1, 2, 3, 4}
• If an element is not a member of (or an element of) a set S, we use the notation a ∉ S - 7 ∉ {1, 2, 3, 4} - Virginia ∉ {1, 2, 3, 4}

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Often used sets

• N = {0, 1, 2, 3, …} is the set of natural numbers
• Z = {…, -2, -1, 0, 1, 2, …} is the set of integers
• Z+^ = {1, 2, 3, …} is the set of positive integers (a.k.a whole numbers) - Note that people disagree on the exact definitions of whole numbers and natural numbers
• Q = { p / q | p ∈ Z , q ∈ Z , q ≠ 0} is the set of rational numbers - Any number that can be expressed as a fraction of two integers (where the bottom one is not zero)
• R is the set of real numbers

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The universal set 2

• For the set of the students in this class, U would be all the students in the University (or perhaps all the people in the world)
• For the set of the vowels of the alphabet, U would be all the letters of the alphabet
• To differentiate U from U (which is a set operation), the universal set is written in a different font (and in bold and italics)

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Venn diagrams

• Represents sets graphically
  • The box represents the universal set
  • Circles represent the set(s)
• Consider set S, which is

the set of all vowels in the alphabet

• The individual elements

are usually not written in a Venn diagram

a e^ i o u

b c^ d f g (^) h j k l m n p q r s t v w x y z

U S

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The empty set 1

• If a set has zero elements, it is called the

empty (or null) set

• Written using the symbol ∅
• Thus, ∅ = { }  VERY IMPORTANT
• If you get confused about the empty set in a problem, try replacing ∅ by { }
• As the empty set is a set, it can be a element

of other sets

• { ∅, 1, 2, 3, x } is a valid set

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The empty set 1

• Note that ∅ ≠ { ∅ }
  • The first is a set of zero elements
  • The second is a set of 1 element (that one element being the empty set)
• Replace ∅ by { }, and you get: { } ≠ { { } }
  • It’s easier to see that they are not equal that way

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Subsets 1

• If all the elements of a set S are also elements of a

set T, then S is a subset of T

• For example, if S = {2, 4, 6} and T = {1, 2, 3, 4, 5, 6, 7}, then S is a subset of T
• This is specified by S ⊆ T
  • Or by {2, 4, 6} ⊆ {1, 2, 3, 4, 5, 6, 7}
• If S is not a subset of T, it is written as such:

S ⊆ T

• For example, {1, 2, 8} ⊆ {1, 2, 3, 4, 5, 6, 7}

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Subsets 2

• Note that any set is a subset of itself!
  • Given set S = {2, 4, 6}, since all the elements of S are elements of S, S is a subset of itself
  • This is kind of like saying 5 is less than or equal to 5
  • Thus, for any set S, S ⊆ S

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• If S is a subset of T, and S is not equal to T,

then S is a proper subset of T

• Let T = {0, 1, 2, 3, 4, 5}
• If S = {1, 2, 3}, S is not equal to T, and S is a subset of T
• A proper subset is written as S ⊂ T
• Let R = {0, 1, 2, 3, 4, 5}. R is equal to T, and thus is a subset (but not a proper subset) or T - Can be written as: R ⊆ T and R ⊄ T (or just R = T)
• Let Q = {4, 5, 6}. Q is neither a subset or T nor a proper subset of T

Proper Subsets 1

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Proper Subsets 2

• The difference between “subset” and “proper

subset” is like the difference between “less

than or equal to” and “less than” for numbers

• The empty set is a proper subset of all sets

other than the empty set (as it is equal to the

empty set)