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SETS
Set = A collection of objects
E
= {0, 2, 4,…}
P = {2, 3, 5, 7,…}
A = {A, B, C, D, . . ., Z}
Two sets are same if their elements are
the same.
A = {1,2,3}
B = {2,1,3}
A
B: If each element of A is in B.
A
B: if A
B and A
B.
A
B,
A
B, A – B.
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View Full Document SOME SPECIAL SETS
: Natural Numbers
: Integers
: Rational Numbers
: Real Numbers
.
P
ROPERTIES
:
S
ET
O
PERATIONS
Let A, B, and C be sets.
Idempotency
A
A = A
A
A
=A
Commutativity
A
B = B
A
A
B = B
A
Associativity
A
(B
C ) = (A
B)
C
A
(B
C ) = (A
B)
C
Distributivity
(A
B)
C = (A
C)
(B
C)
(A
B)
C = (A
C)
(B
C)
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View Full Document Absorption
(A
B)
A = A
(A
B) U A = A
De Morgan’s
A(B
C) = (A – B)
(A C)
A(B
C) = (A – B)
(A  C)
Theorem
A – (B
C) = (A  B)
(A C)
Proof:
Let L = A  (B
C).
Let R =(A – B)
(A – C).
We show L
R and R
L
L = R.
Proof :
L
R
Let x
L.
x
A and x
B
C
x
B or
x
C
x
(A B ) or x
(A – C)
x
(A B )
x
(A – C) = R
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View Full Document Proof:
R
L
Let x
R
x
A  B or
x
A – C
x
A and x
B
C
x
A – ( B
C ) = L
P
ARTITION
The partition of a set A
2
A
such that
1.
Each element of
is
nonempty
.
2.
Distinct members of
are
disjoint
.
3.
= A
Example :
A = {1,2,3,4}
= { {1}, {2,4}, {3}}
Example :
A = N
= {EVEN, ODD}
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View Full Document R
ELATIONS AND
F
UNCTIONS
Cartesian Product of Two Sets
A ×B = Set of all ordered pairs (a, b)
such that a
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This note was uploaded on 11/30/2010 for the course SCS 2805 taught by Professor Smid during the Spring '09 term at Carleton CA.
 Spring '09
 SMID

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