Mathematics for Computer Science
What is a Set?
MIT 6.042J/18.062J
Informally:
A set is a collection of mathematical
objects, with the collection treated
as a single mathematical object.
Sets &
Functions
Albert R Meyer
(This is circular of course:
what’s a collection?)
lec 2F.1
February 12, 2010
Albert R Meyer
February 12, 2010
Some sets
Some sets
R
complex numbers, C
integers,
Z
empty set,
∅
real numbers,
{7, “Albert R.”, π/2, T}
set of all subsets of integers , pow(Z)

 

the power set
Albert R Meyer
A set with 4 elements: two
numbers, a string, and a Boolean.
Same as
{T, “Albert R.”, 7, π/2}
-- order doesn’t matter
February 12, 2010
lec 2F.3
Membership
π/2 ∈ {7, “Albert R.”, π/2, T}
π/3 ∉ {7, “Albert R.”, π/2, T}
14/2 ∈ {7, “Albert R.”, π/2, T}
February 12, 2010
Albert R Meyer
February 12, 2010
lec 2F.4
Synonyms for Membership
x is a member of A: x ∈ A
Albert R Meyer
lec 2F.2
lec 2F.5
x∈A
x is an element of A
x is in A
Examples:
7∈Z,
Albert R Meyer
2/3 ∉ Z, Z ∈pow(R)
February 12, 2010
lec 2F.6
1
In or Not In
Subset (⊆)
An element is in or not in a set:
{7, π/2, 7} is same as {7, π/2}
(No notion of being in the set
more than once)
Albert R Meyer
February 12, 2010
lec 2F.7
A is a subset of B
A is contained in B
Every element of A is also
an element of B:
A⊆B
∀x [x∈A
Albert R Meyer
x∈B]
February 12, 2010
lec 2F.8
∅ ⊆ everything
Subset
def: ∅ ⊆ B
examples:
Z⊆ R, R⊆ C, {3} ⊆ {5,7,3}
A ⊆ A,
IMPLIES
∅ ⊆ every set
Albert R Meyer
February 12, 2010
lec 2F.9
∀x [x∈∅ IMPLIES x∈B]
false true
Albert R Meyer
February 12, 2010
union
New sets from old
A
lec 2F.10
A
B
B
Venn Diagram for 2 Sets
Albert R Meyer
February 12, 2010
lec 2F.14
Albert R Meyer
February 12, 2010
lec 2F.15
2
intersection
A
Albert R Meyer
set difference
A
B
lec 2F.16
February 12, 2010
Albert R Meyer
A set-theoretic equality
proof: Show these have the same
elements, namely,
x∈ Left Hand Set iff x∈ RHS
for all x.
February 12, 2010
lec 2F.18
proof uses fact from last time:
P OR (Q AND R) equiv
(P OR Q) AND (P OR R)
Albert R Meyer
February 12, 2010
lec 2F.19
A set-theoretic equality
A∪(B∩C) = (A∪B)∩(A∪C)
proof: x∈A∪(B∩C)
iff
x∈A OR x∈(B∩C)
(def of ∪) iff
x∈A OR (x∈B AND x∈C) (def ∩) iff
(x∈A OR x∈B) AND (x∈A OR x∈C)
(by the equivalence)
February 12, 2010
lec 2F.17
A∪(B∩C) = (A∪B)∩(A∪C)
A set-theoretic equality
Albert R Meyer
February 12, 2010
A set-theoretic equality
A∪(B∩C) = (A∪B)∩(A∪C)
Albert R Meyer
B
lec 2F.20
proof:
(x∈A OR x∈B)AND(x∈A OR x∈C) iff
(x∈A∪B)AND(x∈A∪C) (def ∪) iff
x ∈(A∪B) ∩ (A∪C) (def ∩).
QED
Albert R Meyer
February 12, 2010
lec 2F.21
3
“is taking subject” relation
subjects
students
Relations &
Functions
is taking
6.042
6.003
6.012
Image by MIT OpenCourseWare.
Albert R Meyer
lec 2F.25
February 12, 2010
formula “evaluation” relation
“nonstop bus trip” relation
arithmetic
formulas
cities
numbers
evaluates to
Albert R Meyer
Feb 17
2
12, 2010
Copyright ©February
Albert
R Meyer
Boston
Boston
sqrt(9)
50/10 – 3
cities
nonstop bus
3
1+2
lec 2F.26
12, 2010
Copyright ©February
Albert
R Meyer
Albert R Meyer
Feb 17
lec 2F.27
Providence
Providence
New York
New York
Albert R Meyer
Binary relations
lec 2F.28
February 12, 2010
Binary relation R from A to B
domain
A
A binary relation, R, from a
set A to a set B
associates of elements of
A with elements of B.
R:
codomain
B
a1
b1
b2
a2
b3
a3
b4
arrows
Albert R Meyer
February 12, 2010
lec 2F.33
Albert R Meyer
Feb 17
February 12, 2010
lec 2F.34
4
Binary relation R from A to B
R:
domain
A
a1
a2
b3
graph(R)
b4
::= the arrows
Albert R Meyer
February 12, 2010
Feb. 17,
A
codomain
B
b1
b2
a3
Binary relation R from A to B
a1
a2
b3
a3
b4
graph(R) = {(a1,b2), (a1,b4), (a3,b4)}
lec 2F.35
≤, ≥ ,= 1 arrow in
A
B
b1
b2
Albert R Meyer
B
February 12, 2010
Feb. 17,
archery on relations
≤, ≥, = 1 arrow out
R:
lec 2F.37
f: A → B
A function, f, from A to B
is a relation which associates
each element, a, of A with
at most one element of B.
called f(a)
Albert R Meyer
Feb. 17, 2009
February 12, 2010
lec 2F.38
Albert R Meyer
function archery
lec 2F.39
function archery
≤ 1 arrow out
≤ 1 arrow out
A
Albert R Meyer
Feb. 17, 2009
February 12, 2010
Feb. 17,
B
February 12, 2010
lec 2F.40
A
Albert R Meyer
Feb. 17, 2009
B
February 12, 2010
lec 2F.41
5
function archery
total relations
≤ 1 arrow out
f( ) =
A
B
Albert R Meyer
Feb. 17, 2009
February 12, 2010
lec 2F.42
R:A→B is total iff
every element of A is
associated with at least
one element of B
Albert R Meyer
Feb. 17, 2009
total relation archery
total relation archery
≥ 1 arrow out
≥ 1 arrow out
A
B
Albert R Meyer
Feb. 17, 2009
February 12, 2010
lec 2F.45
A
B
Albert R Meyer
Feb. 17, 2009
total relation archery
lec 2F.46
February 12, 2010
total & function archery
exactly 1 arrow out
≥ 1 arrow out
f( ) =
A
Albert R Meyer
Feb. 17, 2009
lec 2F.44
February 12, 2010
B
February 12, 2010
lec 2F.47
A
Feb. 17, 2009Albert R Meyer
B
February 12, 2010
lec 2F.49
6
surjections (onto)
surjection archery
≥ 1 arrow in
R:A→B is a surjection
iff every element of B
is associated with some
element of A
Albert R Meyer
Feb. 17, 2009
A
lec 2F.53
February 12, 2010
Albert R Meyer
Feb. 17, 2009
surjection archery
B
surjection archery
≥ 1 arrow in
A
B
Albert R Meyer
Feb. 17, 2009
lec 2F.55
February 12, 2010
≥ 1 arrow in
A
Albert R Meyer
Feb. 17, 2009
surjective & function
≤ 1 arrow out
Albert R Meyer
Feb. 17, 2009
B
injection archery
B
February 12, 2010
lec 2F.56
February 12, 2010
≤ 1 arrow in
≥ 1 arrow in
A
lec 2F.54
February 12, 2010
lec 2F.58
A
Albert R Meyer
Feb. 17, 2009
B
February 12, 2010
lec 2F.62
7
injection archery
injection archery
≤ 1 arrow in
A
Albert R Meyer
Feb. 17, 2009
≤ 1 arrow in
B
February 12, 2010
lec 2F.63
A
B
Albert R Meyer
Feb. 17, 2009
bijection archery
exactly 1 arrow out
Albert R Meyer
Feb. 17, 2009
B
Copyright © Albert R.February
Meyer,12,
2009.
2010 All rights reserved.
lec 2F.64
Team Problems
exactly 1 arrow in
A
February 12, 2010
lec 2F.69
Problems
1―3
Albert R Meyer
February 12, 2010
lec 2F.71
8
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6.042J / 18.062J Mathematics for Computer Science
Spring 2010
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