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

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Set Operations

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• Triangle shows mixable color range (gamut) – the set of colors

Sets of Colors

Monitor gamut (M)

Printer gamut (P)

• Pick any 3 “primary” colors

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Set operations: Union 2

U

A B

A U B

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Set operations: Union 3

• Formal definition for the union of two sets:

A U B = { x | x ∈ A or x ∈ B }

• Further examples

• {1, 2, 3} U {3, 4, 5} = {1, 2, 3, 4, 5}
• {New York, Washington} U {3, 4} = {New

York, Washington, 3, 4}

• {1, 2} U ∅ = {1, 2}

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• An intersection of the sets contains all the elements in BOTH sets - Intersection symbol is a ∩ - Example: C = M ∩ P

Monitor gamut (M)

Printer gamut (P)

Set operations: Intersection 1

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Set operations: Intersection 2

U

A B

A ∩ B

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Set operations: Intersection 4

• Properties of the intersection operation

• A ∩ U = A Identity law
• A ∩ ∅ = ∅ Domination law
• A ∩ A = A Idempotent law
• A ∩ B = B ∩ A Commutative law
• A ∩ (B ∩ C) = (A ∩ B) ∩ C Associative law

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Disjoint sets 1

• Two sets are disjoint if the have NO elements in common
• Formally, two sets are disjoint if their intersection is the empty set - Another example: the set of the even numbers and the set of the odd numbers

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Disjoint sets 3

• Formal definition for disjoint sets: two sets

are disjoint if their intersection is the empty

set

• Further examples

• {1, 2, 3} and {3, 4, 5} are not disjoint
• {New York, Washington} and {3, 4} are

disjoint

• {1, 2} and ∅ are disjoint
  • Their intersection is the empty set
• ∅ and ∅ are disjoint!
  • Their intersection is the empty set

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Set operations: Difference 1

• A difference of two sets is the elements in one set that are NOT in the other - Difference symbol is a minus sign - Example: C = M - P

Monitor gamut (M)

Printer gamut (P)

• Also visa-versa: C = P - M

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• Formal definition for the difference of two

sets:

A - B = { x | x ∈ A and x ∉ B }

A - B = A ∩ B  Important!

• Further examples

• {1, 2, 3} - {3, 4, 5} = {1, 2}
• {New York, Washington} - {3, 4} = {New York,

Washington}

• {1, 2} - ∅ = {1, 2}
  • The difference of any set S with the empty set will be the set S

Set operations: Difference 3

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• A symmetric difference of the sets contains all the elements in either set but NOT both - Symetric diff. symbol is a ⊕ - Example: C = M ⊕ P

Monitor gamut (M)

Printer gamut (P)

Set operations: Symmetric

Difference 1

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• A complement of a set is all the elements that are NOT in the set - Difference symbol is a bar above the set name: P or M

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Monitor gamut (M)

Printer gamut (P)

Complement sets 1

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Complement sets 2

U

A

A

B

B

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