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## Sample Document Text

Sets
Definition of a Set:
NAME = {list of elements or description of elements}
i.e. B = {1,2,3} or C = {x?Z
+
| -4 < x < 4}
Axiom of Extension:
A set is completely defined by its elements
i.e. {a,b} = {b,a} = {a,b,a} = {a,a,a,b,b,b}
Subset
A?B ??x?U, x?A?x?B
A is contained in B
B contains A
A? B ??x ?U, x?A ^ x?B
Relationship between membership and subset:
?x?U, x?A ? {x} ? A
Definition of set equality: A = B ? A? B ^ B? A
Same Set or Not??
X={x?Z | ?p ?Z, x = 2p}
Y={y?Z | ?q?Z, y = 2q-2}
A={x?Z | ?i ?Z, x = 2i+1}
B={x?Z | ?i ?Z, x = 3i+1}
C={x?Z | ?i ?Z, x = 4i+1}
Set Operations
Formal Definitions and Venn Diagrams
Union:
Intersection:
Complement:
Difference:
}|{ BxAxUxBA ????=U
}|{ BxAxUxBA ????=I
}|{ BxAxUxBA ????=?
}|{' AxUxAA
c
??==
'BABA I=?
Ordered n-tuple
and the Cartesian Product
. Ordered n-tuple - takes order and multiplicity into account
.(x
1
,x
2
,x
3
,.,x
n
)
- n values
- not necessarily distinct
- in the order given
.(x
1
,x
2
,...

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