Sets

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Sets
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Set builder notation

• N 0,1,2,3, the natural numbers
• R reals
• Z -3,-2,-1,0,1,2,3,
• Z 1,2,3,4,5,
• Q the set of rational numbers

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Set builder notation
S contains all elements from U (universal
set) That make predicate P true
Brace notation with ellipses (wee dots)
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Set Membership
x is in the set S (x is a member of S)
y is not in the set S (not a member)
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Am I making this up?
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Set operators
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Set empty
The empty set
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The universal set
U
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Venn Diagram
U
B
A
U is the universal set A is a subset of B
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Venn Diagram
U
B
A
U is the universal set A united with B
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Venn Diagram
U
B
A
U is the universal set A intersection B
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Cardinality of a set
The number of elements in a set (the size of the
set)
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Power Set
The set of all possible sets
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Power Set
The set of all possible sets
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Power Set
The set of all possible sets
We could represent a set with a bit string
0th element
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Is this true for a set S?
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Cartesian Product
A set of ordered tuples
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(improper) Subset

• is empty a subset of anything?
• Is anything a subset of ?
• We have an implication, what is its truth table?
• Note improper subset!!

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(proper) Subset
Consequently A is strictly smaller than B A lt
B
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Equal sets?
Two show that 2 sets A and B are equal we need to
show that
And we know that
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Try This

• Using set builder notation describe the following
  sets
• odd integers in the range 1 to 9
• the integers 1,4,9,16,25
• even numbers in the range -8 to 8

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Answers

• Using set builder notation describe the following
  sets
• odd integers in the range 1 to 9
• the integers 1,4,9,16,25
• even numbers in the range -8 to 8

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Go do this in claire

• build the sets we just mentioned
• test if is a subset of itself

• Using set builder notation describe the following
  sets
• odd integers in the range 1 to 9
• the integers 1,4,9,16,25
• even numbers in the range -8 to 8

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How might a computer represent a set?
Remember those bit operations?
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Computer Representation (possible

• How do we compute the following?
• membership of an element in a set
• union of 2 sets
• intersection of 2 sets
• compliment of a set
• set difference (tricky?)

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Power set
Try this

• Compute the power set of
• 1,2
• 1,2,3
• 1,2

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Power set

• Compute the power set of
• 1,2
• 1,2,3
• 1,2

Do it in claire
Think again how might we represent sets?
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Cartesian Product

• A set of ordered tuples
• note AxB is not equal to BxA

Just reminding you (and me)
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Try This

• Let A1,2,3 and Bx,y, find
• AxB
• BxA
• if An and Bm what is AxB

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My answer
Do it in claire

• Let A1,2,3 and Bx,y, find
• AxB
• BxA
• if An and Bm what is AxB

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For the brave the claire code
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member(eany,Aset) boolean -gt exists(x in A
x e) // // There exists some element x in A
such that // x e // subset(Aset,Bset)
boolean -gt forall(x in A member(x,B)) // //
All elements of A are in B. Sometimes called
"improper" subset // What does it do when A ?
Hint P -gt Q and P is false! //
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PS(Aset) set -gt PS(A,) PS(A,Bset)
set -gt set(B) PS(Aset,Bset) void -gt let
x A1 in PS(delete(copy(A),x),add(copy(B),
x)) U PS(delete(copy(A),x),B) // // The
power set of A // NOTE U is the (claire) union
operator //
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CP(Aset,Bset) set -gt let pairs
in (for x in A (for y in B pairs
add(pairs,list(x,y))), pairs) // // The
cartesian product of 2 sets A and B // Produce a
set of tuples (as a list) // // Demo
CP(1,2,3,1,5) // CP(1,2,3,"A","B") /
/