Relations Between Sets
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1
Relations
Students
Courses
Sam
EC
10
Mary
CS20
The “is-taking” relation
A relation is a set of ordered pairs:
{(Sam,Ec10), (Sam, CS20), (Mary, CS20)}
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Function: A → B
AT MOST ONE ARROW OUT OF EACH ELEMENT OF A
domain
A
f
codomain
B
Each element of A is associated with at most one element of B.
⟼b
f(a) = b
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a
3
Total Function: A → B
EXACTLY ONE ARROW OUT OF EACH ELEMENT OF A
domain
A
f
codomain
B
Each element of A is associated with ONE AND ONLY one element of B.
a⟼b
f(a) = b
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A Function that is “Partial,”
Not Total
domain
R×R
f
codomain
R
f: R ×R → R
f(x,y) = x/y
Defined for all pairs (x,y) except when y=0!
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A Function that is “Partial,”
Not Total
domain
R×R
f
codomain
R
f: R ×R → R
f(x,y) = x/y
Defined for all pairs (x,y) except when y=0!
Or: f is a total function: R×(R-{0})→R
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Injective Function
“at most one arrow in”
domain
f
A
codomain
B
(∀b∈B)(∃≤1a∈A) f(a)=b
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Surjective Function
“at least one arrow in”
domain
f
A
codomain
B
(∀b∈B)(∃≥1a∈A) f(a)=b
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Bijection =
Total + Injective + Surjective
“exactly one arrow out of each element of A
and exactly one arrow in to each element of B”
domain
f
A
2/13/12
codomain
B
(∀a∈A) f(a) is defined and
(∀b∈B)(∃=1a∈A) f(a)=b
9
Cardinality or “Size”
For finite sets, a bijection exists iff A and B have
the same number of elements
domain
A
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f
codomain
B
10
Cardinality or “Size”
Use the same as a definition of “same size” for
infinite sets:
Sets A and B have the same size iff there is a
bijection between A and B
Theorem: The set of even integers has the
same size as the set of all integers [f(2n) = n]
…, -4, -3, -2, -1, 0, 1, 2, 3, 4 …
…, -8, -6, -4, -2, 0, 2, 4, 6, 8 …
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Cardinality or “Size”
There are as many natural numbers as integers
0 1 2 3 4 5 6 7 8…
0, -1, 1, -2, 2, -3, 3, -4, 4 …
f(n) = n/2 if n is even, -(n+1)/2 if n is odd
Defn: A set is countably infinite if it has the
same size as the set of natural numbers
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An Infinite Set May Have the Same
Size as a Proper Subset!
⋮
⋮
5
5
4
4
3
3
2
2
1
1
0
0
Hilton
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Every room of both
hotels is full!
Suppose the Sheraton
has to be evacuated
Sheraton
13
An Infinite Set May Have the Same
Size as a Proper Subset!
⋮
⋮
5
5
4
4
3
3
2
2
1
1
0
0
Hilton
2/13/12
Step 1: Tell the resident
of room n in the Hilton
to go to room 2n
This leaves all the oddnumbered rooms of the
Hilton unoccupied
Sheraton
14
An Infinite Set May Have the Same
Size as a Proper Subset!
⋮
⋮
5
5
4
4
3
3
2
2
Step 2: Tell the resident
of room n in the
Sheraton to go to room
2n+1 of the Hilton.
1
1
Everyone gets a room!
0
0
Hilton
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Sheraton
15