2.1 – sets. examples: set-builder notation using set-builder notation to make domains explicit...
TRANSCRIPT
2.1 – Sets Definition:
Notation: • Uppercase letters A, B, S, T, etc.• Set braces, e.g. S={2, red, water, {1}}• Membership:
means means
• {2,2,4,5,4,7,2}=
Examples:
• Let S be the set of all of the people in this room.
• Standard sets:
Set-Builder Notation
• General form is { x | P(x) } for some predicate P(x).
• Examples:
• Symbol | or : in the notation means
Using Set-Builder Notation to Make Domains Explicit
• Examples
}
{𝑥∈ℤ∨𝑥2<10 }
{𝑥∈ℕ∨𝑥2<10 }
Examples:
(a) Set of all integers which are perfect squares.
(b) {2,4,6}
Venn Diagrams
• “Universal Set” U
• Picturing set as a restricted portion of the universal set
Special Kinds of Sets
• Empty set
• Question:
Subsets
• Notation • Define using the predicate calculus:
• Questions: For all sets S,.– Is ?– Is ?
Is ?
Is ?
Set Equality
if and only if
•
•
•
Miscellaneous
• Proper subsets
• Cardinality of a set
• Finite sets
• Infinite sets
New Sets from Old
• The power set
• Cartesian products (ordered pairs, n-tuples)
Examples: 𝐵={1,2 } ,𝐶= {𝑎 ,𝑏 ,𝑐 }
𝐵×𝐶=¿
|𝐵×𝐶|=¿
𝑃 (∅ )=¿
𝑃 ( {∅ } )=¿
𝑃 (𝐵 )=¿
2.2 Set Operations
• The union of two sets and is the set of all elements which are either in or in .
• Set-Builder notation:
• Venn Diagram:
Intersection
• The intersection of two sets and is the set of all elements common to both.
• Set-builder notation:
• Venn Diagram:
Generalized Unions and Intersections
• These are well-defined because of associativity
More Definitions
• Disjoint sets:
• Mutually Disjoint collections of sets:
More Definitions
• Principle of Inclusion-Exclusion
• Set Difference
Set Complement
• Definition of set complement:
• Venn Diagram:
Proving Set Identities (Listed in Table 1 on page 124)
𝐴∩ ( 𝐴∪𝐵 )=𝐴 (2nd Absorption Law)
𝐴∪𝐵=𝐴∩𝐵 1st De Morgan Law