Set Theory

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Set Theory
A
B
C

• Lecture 10

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Sets
Informally A set is a collection of
mathematical objects, with the collection
treated as a single mathematical object.
Examples

• real numbers, ?
• complex numbers, C
• integers, ?
• empty set, ?

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Defining Sets
Sets can be defined directly
e.g. 1,2,4,8,16,32,,
CSC1130,CSC2110,
Order, number of occurence are not important.
e.g. A,B,C C,B,A A,A,B,C,B
A set can be an element of another set.
1,2,3,4
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Defining Sets by Predicates
The set of elements, x, in A such that P(x) is
true.
The set of prime numbers
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Membership
7, Albert, ?/2, T
x is an element of A x is in A
Examples
?/2 ? 7, Albert,?/2, T ?/3 ? 7,
Albert,?/2, T 14/2 ? 7, Albert,?/2, T
7? ?
2/3 ? ?
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Containment
A is a subset of B A is contained in B
Every element of A is also an element of B.
Examples ??R, 3?5,7,3 ? ? every set,
A ? A
A is a proper subset of B
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Some Examples
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Basic Operations on Sets
union
intersection
difference
complement
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Some Examples
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Partitions of Sets
Two sets are disjoint if their intersection is
empty.
A collection of nonempty sets A1, A2, , An is
a partition of a set A if and only if
A1, A2, , An are mutually disjoint.
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Power Sets
power set
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Set Identities
Distributive Law
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Set Identities
De Morgans Law
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Proving Set Identities
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Proving Set Identities
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Russells Paradox
so
There is a male barber who shaves all those men,
and only those men, who do not shave themselves.
Does the barber shave himself?
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Halting Problem
We want a program H that given any program P and
input I H(P,I) returns halt if P will
terminate given input I H(P,I) returns loop
forever if P will not terminate given input I.
No such program can terminate in a finite number
of steps!
(page 295 of the textbook)