CSE115/ENGR160 Discrete Mathematics
02/14/12
Ming-Hsuan Yang
UC Merced
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2.2 Set operations
• Union: the set that contains those elements
that are either in A or in B, or in both
A  B  {x | x  A  x  B}
• A={1,3,5}, B={1,2,3}, A⋃B={1,2,3,5}
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Intersection
• Intersection: the set containing the elements
in both A and B
A  B  {x | x  A  x  B}
• A={1,3,5}, B={1,2,3}, A⋂B={1,3}
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Disjoint set
• Two sets are disjoint if their intersection is ∅
• A={1,3}, B={2,4}, A and B are disjoint
• Cardinality: | A  B || A |  | B |  | A  B |
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Difference and complement
• A-B: the set containing those elements in A
but not in B A  B  {x | x  A  x  B}
• A={1,3,5},B={1,2,3}, A-B={5}
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Complement
• Once the universal set U is specified, the
complement of a set can be defined
• Complement of A: A  {x | x  A}, A  U  A
• A-B is also called the complement of B with
respect to A
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Example
• A is the set of positive integers > 10 and the
universal set is the set of all positive integers,
then A  {x | x  10}  {1,2,3,4,5,6,7,8,9,10}
• A-B is also called the complement of B with
respect to A
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Set identities
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Example
• Prove A  B  A  B
• Will show that A  B  A  B and A  B  A  B
• (→): Suppose that x  A  B , by definition of
complement and use De Morgan’s law
( x  A  x  B)
 (( x  A))  (( x  B))
 ( x  A)  ( x  B)
• By definition of complement x  A or x  B
• By definition of union x  A   B
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Example
•
•
•
•
•
(←): Suppose that x  A  B
By definition of union x  A  x  B
By definition of complement x  A  x  B
Thus ( x  A)  ( x  B)
By De Morgan’s law: ( x  A)  ( x  B)
 ( x  A  x  B)
 ( x  ( A  B))
• By definition of complement, x  A  B
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Builder notation
• Prove it with builder notation
A  B  {x | x  A  B} (def of complement)
 {x | ( x  ( A  B))} (def of not belong t o)
 {x | ( x  A  x  B)} (def of int ersecton)
i
 {x | ( x  A)  ( x  B )} (De Morgan's law)
 {x | x  A  x  B} (def of not belong t o)
 {x | x  A  x  B} (def of complement)
 {x | x  A  B} (def of union)
 A B
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Example
• Prove A  ( B  C )  ( A  B)  ( A  C )
• (→): Suppose that x  A  ( B  C ) then x  A
and x  B  C . By definition of union, it
follows that x  A , and x  B or x  C .
Consequently, x  A and x  B or x  A and x  C
• By definition of intersection, it follows x  A  B
or x  A  C
• By definition of union, x  ( A  B)  ( A  C )
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Example
• (←): Suppose that x  ( A  B)  ( A  C )
• By definition of union, x  A  B or x  A  C
• By definition of intersection, x  A and x  B ,
or x  A and x  C
• From this, we see x  A , and x  B or x  C
• By definition of union, x  A and x  B  C
• By definition of intersection, x  A  ( B  C )
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Membership table
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Example
• Show that
A  (B  C)  (C  B)  A
A  ( B  C )  A  B  C (De Morgan's law)
 A  ( B  C ) (De Morgan's law)
 ( B  C )  A (commutati
ve law)
 (C  B)  A (commutati
ve law)
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Generalized union and intersection
• A={0,2,4,6,8}, B={0,1,2,3,4}, C={0,3,6,9}
• A⋃B⋃C={0,1,2,3,4,6,8,9}
• A⋂B⋂C={0}
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General case
•
•
•
•
•
n
Union: A  A   A   A
Intersection A  A   A   A
Union: A  A    A     A
Intersection: A  A    A     A
Suppose Ai={1, 2, 3,…, i} for i=1,2,3,…
1
2
n
1
i
i 1
n
2
n
i 1

1
2
n
1


i 1
i 1


i
i 1
2
i

n
i 1
i

A

{
1
,
2
,
3
,

,
i
}

{
1
,
2
,
3
,

}

Z
i
 
 A  {1,2,3,, i}  {1}
i 1
i
i 1
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Computer representation of sets
• U={1,2,3,4,5,6,7,8,9,10}
• A={1,3,5,7,9} (odd integer ≤10),B={1,2,3,4,5}
(integer ≤5)
• Represent A and B as 1010101010, and
1111100000
• Complement of A: 0101010101
• A⋂B: 1010101010˄1111100000=1010100000
which corresponds to {1,3,5}
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