22C:19 Discrete Math Sets and Functions Fall 2011 Sukumar Ghosh.
Discrete Mathematics Ch. 5 Sets (Review)
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Transcript of Discrete Mathematics Ch. 5 Sets (Review)
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Discrete Mathematics
Ch. 5 Sets (Review)
Instructor: Hayk [email protected]
Today we will review sections 5.1, 5.2, 5.3
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What is a set?
A collection of elements:Order is irrelevantNo repetitionsCan be infiniteCan be empty
Examples:{Angela, Belinda, Jean}{0,1,2,3,…}
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Operations on sets
S is a set Membership:xS x is an element of S Angela{Angela, Belinda, Jean}
Subset S1 S – Set S1 is a subset of set S– All elements of S1 are elements of S– {Angela,Belinda} {Angela, Belinda, Jean}
Proper subset S1 S
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Operations on sets
If S, S1 are sets
Intersection: S S1
– is a set of all elements that belong to both
{Ang, Bel, Jea} {Ang, Dan} = {Ang}
Union: S S1
– is a set of all elements that belong to either
– {Ang, Bel, Jea} {Ang, Dan} = {Ang, Bel, Jea, Dan}
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Operations on sets
Let S, S1 be sets
Equality: S = S1
– iff they have the same elements
Difference: S \ S1
– is a set of all elements that belong to S but NOT to S1
{Ang, Bel, Jea} \ {Ang, Dan} = {Bel, Jea}
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More notation
In mathematics sets are often specified with a predicate and an enveloping set as follows:
S = {xA | P(x)}
S is the set of all elements of A that satisfy predicate P
Example:
Q={xR | a,bZ b0 & x=a/b}
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Set Equality
Two sets are equal iff they have the same elements
Theorem: for any sets A and B, A=B iff AB & BA
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Book example 5.1.5
– 2{1,2,3} ?– {2}{1,2,3} ?– 2{1,2,3} ?– {2}{1,2,3} ?– {2}{{1},{2}} ?– {2}{{1},{2}} ?
How about set A such that {2} is a subset of it and A is an element of it?– A={1,2,{1},{2}}
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Universal SetIf we are dealing with sets which are all
subsets ofa larger set U then we call it a universal set U
All of your sets will be subsets of U
When does such a U exist?
Always, for we can set U to the union of all sets involved ???????
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Complement
So if I am dealing with set A which is a subset of the universal set U then:
I can define complement of A: AC = U\AThat is the set of all elements (of U) that are not in AOften “of U” is dropped and people say that AC isthe set of everything that is not in A
What is the complement of U? UC = Ø
What set has U as its complement? ØC=U
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Sets & Predicate LogicAll of the set operations and relations above
can be defined in terms of Boolean connectives:– AB={x | xA v xB}– AB={x | xA & xB}– A\B={x | xA & not xB}– AC={x | not xA}
– A = B iff x( xA xB)– A B iff x (xA xB)– AB iff x (xA xB) & not A=B
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Symmetric Difference
Set C is the symmetric difference of sets A and
B iff every element of C belongs to A or B butnot both
ABC [C=A B a (aC (aA xor aB))]
If A={1,2}, B={2,3} then A B={1,3}
In general: A A = {}
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Exercise 2Intersection of two sets is contained in their
union: AB [ (A B) (A B) ]Proof:
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Exercise 3
Union is commutativeAB [ A B = B A ]
Intersection is commutative
AB [ A B = B A ]
Intersection distributes over union:
ABC [ A (B C) =(A B) (A C) ]
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Exercise
In Exercise #5 we proved: ABC [ A (B C) =(A B) (A C) ]
using the fact that A&(BvC) = (A&B)v(A&C)Given the statement just proved
Av(B&C) = (AvB) & (AvC)
what can we now prove in terms of sets?Union distributes over intersection:
ABC [ A (B C) =(A B) (A C) ]
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Proof :
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Logic - Sets
v & avb=bva ab=ba a&b=b&a ab=ba (avb)vc=av(bvc) (ab)c=a(bc) (a&b)&c=a&(b&c) (ab)c=a(bc) a&(bvc)=(a&b)v(a&c)
a(bc)=(ab)(ac) av(b&c)=(avb) & (avc)
a(bc)=(ab)(ac)
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Cartesian Products
Intuition first:Suppose I have a function that takes two
numbers x and y and returns x/yWhat is the set of valid inputs?Is it just R?
– No -- cannot divide by 0
Is it R\{0}?– No -- can happily have 0 as x
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CombinationsSuppose I have:• two independent attributes:
sky conditions and temperaturetwo values for the sky conditions
S={sunny, overcast};three values for the precipitation:
P={snow, rain, nothing}.
How many combinations can I have?
<sunny, rain> , <sunny, snow> . <sunny, nothing>.
<overcast, rain>. <overcast, snow>, <overcast, nothing>
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Cartesian Product
Set C is a Cartesian Product of set A and set B iff it is a set of all ordered pairs such that the 1st element belongs to A and the 2nd element belongs to B
C=A B iff ab (<a, b>C (aA & bB))]
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Examples
A={0,1}, B={Ang, Bel}AB = {<0,Ang>, <0,Bel>, <1,Ang>, <1,
Bel>}
A={0,1}, B={Ang, Bel}BA = {<Ang, 0>, <Bel, 0>, <Ang, 1>,
<Bel,1>}
A={0}, B={a,b}, C={1,2}ABC={<0, a, 1>,<0, b, 1>, <0, a, 2>,<0,
b, 2>}
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More examples
A=B=C=D=R (set of all real numbers)ABCD=R4 (time-space continuum)
What is the cardinality of Cartesian Product?
|A1 … An|=|A1| · … · |An| for finite sets
How about { } {1,2}?– {} {1,2}={}