set operations. when sets are equal a equals b iff for all x, x is in a iff x is in b or … and...
TRANSCRIPT
Set Operations
When sets are equal
][ BxAxxBA
A equals B iff for all x, x is in A iff x is in B
)]()[( AxBxBxAxxBA
or
ABBABA
… and this is what we do to prove sets equal
Note: remember all that stuff about implication?
tivecontraposi
inverse
converse
nalbiconditio
qppq
qp
pq
pqqpqp
}|{ BxAxxBA
Union of two sets
Give me the set of elements, x where x is in A or x is in B
}5,4,3,2,1{A }8,7,6,5,4{B
}8,7,6,5,4,3,2,1{BA
Example
BABA0 0 00 1 11 0 11 1 1 OR
A membership table
Union of two sets
}|{ BxAxxBA
Note: we are using set builder notation and the
laws of logical equivalence (propositional equivalence)
}|{ BxAxxBA
Intersection of two sets
Give me the set of elements, x where x is in A and x is in B
}5,4,3,2,1{A }8,7,6,5,4{B
}5,4{ BA
Example
BABA0 0 00 1 01 0 01 1 1
AND
}|{ BxAxxBA
),(disjoint{} BABA
Disjoint sets
][ BxAxx
}|{ BxAxxBA
Difference of two sets
Give me the set of elements, x where x is in A and x is not in B
}5,4,3,2,1{A }8,7,6,5,4{B
}3,2,1{ BA
Example
BA BA0 0 00 1 01 0 11 1 0 BA
BABA
}|{ BxAxxBA
Note: Compliment of a set
}|{ AxUxxA
)()( ABBABA
Symmetric Difference of two sets
Give me the set of elements, x where x is in A and x is not in B OR x is in B and x is not in A
}5,4,3,2,1{A }8,7,6,5,4{B
}8,7,6,3,2,1{BA
Example
BABA0 0 00 1 11 0 11 1 0
XOR
)()( ABBABA
}|{ AxxA
Complement of a set
Give me the set of elements, x where x is not in A
}5,4,3,2,1{A
}10,9,8,7,6,5,4,3,2,1,0{U
}10,9,8,7,6,0{A
Example
Not
U is the “universal set”
AUA
Cardinality of a Set
In claire
• A = {1,3,5,7}• B = {2,4,6}• C = {5,6,7,8}
• |A u B| ?• |A u C|• |B u C|• |A u B u C|
Cardinality of a Set
|BA| |B| |A| || BA
The principle of inclusion-exclusion
Cardinality of a Set
|BA| |B| |A| || BA
The principle of inclusion-exclusion
U
Potentially counted twice (“over counted”)
Set Identities
{} AA
AUA Identity
UUA {}{} A
Domination
•Think of• U as true (universal)• {} as false (empty)• Union as OR• Intersection as AND• Complement as NOT
AAA AAA
Indempotent
Note similarity to logical equivalences!
Set Identities
ABBA ABBA
Commutative
CBACBA )()(CBACBA )()(
Associative
)()()( CABACBA )()()( CABACBA
Distributive
BABA
BABA De Morgan
Note similarity to logical equivalences!
lawsnegation
laws absorption)(
)(
laws sMorgan' De)(
)(
law vedistributi)()()(
)()()(
laws eassociativ)()(
)()(
laws ecommutativ
lawnegation double)(
lawst indempoten
law domination
lawidentity
NameeEquivalenc
Fpp
Tpppqpp
pqppqpqp
qpqprpqprqp
rpqprqprqprqp
rqprqppqqp
pqqp
pp
ppp
pppFFp
TTppFp
pTp
Four ways to prove two sets A and B equal
• a membership table• a containment proof
• show that A is a subset of B• show that B is a subset of A
• set builder notation and logical equivalences• Venn diagrams
B BA :RTP A
Prove lhs is a subset of rhs
Prove rhs is a subset of lhs
B BA :RTP A
… set builder notation and logical equivalences
}|{ BAxx
)}(|{ BAxx
)}(|{ BxAxx
)}()(|{ BxAxx
}|{ BxAxx }|{ BAxx
Defn of complement
Defn of intersection
De Morgan law
Defn of complement
Defn of union
B BA :RTP A
prove using membership table
Class
B BA :RTP AMe
0000111
1101001
1011010
1111000
BABABABABA
They are the same
B BA :RTP A
prove using set builder and logical equivalence
Class
B BA :RTP AMe
notationbuilder set of Meaning
ion)(intersect ofDefn }|{
complement ofDefn )}()(|{
ofDefn )}()(|{
lawMorgan De ))}()(|{
)( ofDefn }(|{
ofDefn )}(|{
}|{
BA
BAxx
BxAxx
BxAxx
BxAxx
unionBxAxx
BAxx
BAxxBA
{})( :RTP ABA
)}(|{)(
ABxAxxABA
)}(|{ AxBxAxx
}|{ BxAxAxx
}{}|{ Bxxx
{}
Prove using set builder and logical equivalences
A containment proof
See the text book
That’s a cop out if ever I saw one!
A containment proof Guilt kicks in
To do a containment proof of A = B do as follows
1. Argue that an arbitrary element in A is in B i.e. that A is an improper subset of B
2. Argue that an arbitrary element in B is in A i.e. that B is an improper subset of A
3. Conclude by saying that since A is a subset of B, and vice versa then the two sets must be equal
ni
n
i
AAAA
...211
Collections of sets
ni
n
i
AAAA
...211
