9/19/10
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Lecture 8
CS336
f10
Inductively Defined
Sets
What We’ll Discuss
Inductively defined sets
Inductively defined functions
Proofs by Mathematical Induction
Specifying sets by
Induction
•
Sets can be specified by:
•
enumeration (useful only for small sets)
•
a defining property —a predicate— that filters out
some elements of some previously defined (larger) set.
•
A third —and important— method is
inductive
construction.
Inductive definitions of sets have
three clauses.
•
Letting S be the set being defined, these are…
–
Basis: a0,a1,…,an
∈
S.
Establishes that S≠
∅
, and characterizes a finite set of “atoms”
from which the set’s other elements are constructed.
Inductive definitions of sets have
three clauses.
•
Letting S be the set being defined, these are…
–
Basis: a0,a1,…,an
∈
S.
Establishes that S≠
∅
, and characterizes a finite set of “atoms”
from which the set’s other elements are constructed.
–
Induction: if x0,…,xm
∈
S then f(x0,…,xm)
∈
S.
The function f specifies how to combine elements of S to form
new elements of S; it is known as the set’s constructor.
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 Fall '08
 Myers
 Natural number, Inductively defined sets

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