CSE115/ENGR160 Discrete Mathematics 02/14/12
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Transcript of CSE115/ENGR160 Discrete Mathematics 02/14/12
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CSE115/ENGR160 Discrete Mathematics02/14/12
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={1,3,5}, B={1,2,3}, A⋃B={1,2,3,5}2
}|{ BxAxxBA
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Intersection
• Intersection: the set containing the elements in both A and B
• A={1,3,5}, B={1,2,3}, A⋂B={1,3}3
}|{ BxAxxBA
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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:
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|||||||| BABABA
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Difference and complement
• A-B: the set containing those elements in A but not in B
• A={1,3,5},B={1,2,3}, A-B={5}5
}|{ BxAxxBA
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Complement
• Once the universal set U is specified, the complement of a set can be defined
• Complement of A: • A-B is also called the complement of B with
respect to A
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AUAAxxA },|{
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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-B is also called the complement of B with respect to A
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}10,9,8,7,6,5,4,3,2,1{}10|{ xxA
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Set identities
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Example
• Prove• Will show that• (→): Suppose that , by definition of
complement and use De Morgan’s law
• By definition of complement• By definition of union
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BABA BABABABA and
BAx
)()(
))(())((
)(
BxAx
BxAx
BxAx
BxAx or
BAx
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Example
• (←): Suppose that• By definition of union• By definition of complement• Thus• By De Morgan’s law:
• By definition of complement, 10
BAx
BxAx
BxAx )()( BxAx
))((
)(
)()(
BAx
BxAx
BxAx
BAx
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Builder notation
• Prove it with builder notation
11BA
BAxx
BxAxx
BxAxx
BxAxx
BxAx
BAxx
BAxxBA
union) of (def }|{
)complement of (def }|{
to)belongnot of (def }|{
law) sMorgan' (De )}()(|{
on)intersecti of (def )}(|{x
to)belongnot of (def ))}((|{
)complement of (def }|{
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Example
• Prove• (→): Suppose that then and . By definition of union, it follows
that , and or . Consequently, and or and
• By definition of intersection, it follows or• By definition of union,
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)()()( CABACBA
)( CBAx AxCBx
Ax Bx CxAx Bx Ax Cx
BAx CAx
)()( CABAx
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Example
• (←): Suppose that • By definition of union, or • By definition of intersection, and ,
or and • From this, we see , and or• By definition of union, and• By definition of intersection,
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)()( CABAx
BAx CAx
Ax BxAx Cx
Ax Bx Cx
Ax CBx )( CBAx
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Membership table
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Example
• Show that
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ABCCBA )()(
law) ve(commutati )(
law) ve(commutati )(
law) sMorgan' (De )(
law) sMorgan' (De )(
ABC
ACB
CBA
CBACBA
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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
• Union:• Intersection• Union:• Intersection:• Suppose Ai={1, 2, 3,…, i} for i=1,2,3,…
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i
n
in AAAA
1
21
i
n
in AAAA
1
21
ii
n AAAA
1
21
ii
n AAAA
1
21
}1{},,3,2,1{
},3,2,1{},,3,2,1{
11
11
iA
ZiA
ii
i
ii
i
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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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