Lecture 2.1: Sets and Set Operations
CS 250, Discrete Structures, Fall 2015
Nitesh Saxena
Adopted from previous lectures by Cinda Heeren
Course Admin
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HW1 Due
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11am 09/17/14 – this Thursday
Please follow all instructions
Recall: late submissions will not be accepted
Travel next week
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Attending and presenting at a conference in
Vienna:
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http://esorics2015.sba-research.org/
No class next week (Tuesday and Thursday)
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7/13/2017
Would not affect our coverage
Please utilize this time to review the previous lectures
Lecture 2.1 -- Sets and Set
Operations
Outline
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Set Definitions and Theory
Set Operations
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Lecture 2.1 -- Sets and Set
Operations
Set Theory - Definitions and notation
A set is an unordered collection of elements.
Some examples:
{1, 2, 3} is the set containing “1” and “2” and “3.”
{1, 1, 2, 3, 3} = {1, 2, 3} since repetition is irrelevant.
{1, 2, 3} = {3, 2, 1} since sets are unordered.
{1, 2, 3, …} is a way we denote an infinite set (in this case,
the natural numbers).
Note:   {}
 = {} is the empty set or null set, or the set containing no
elements.
U: is the set of all possible elements in the universe
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Lecture 2.1 -- Sets and Set
Operations
Set Theory - Definitions and notation
x  S means “x is an element of set S.”
x  S means “x is not an element of set S.”
A  B means “A is a subset of B.”
or, “B contains A.”
or, “every element of A is also in B.”
or, x ((x  A)  (x  B)).
A
B
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Venn Diagram
Lecture 2.1 -- Sets and Set
Operations
Set Theory - Definitions and notation
A  B means “A is a subset of B.”
A  B means “A is a superset of B.”
A = B if and only if A and B have exactly
the same elements.
iff, A  B and B  A
iff, A  B and A  B
iff, x ((x  A)  (x  B)).
So to show equality of sets A and B,
show:
•
•
AB
BA
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Lecture 2.1 -- Sets and Set
Operations
Set Theory - Definitions and notation
A  B means “A is a proper subset of B.”
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A  B, and A  B.
x ((x  A)  (x  B))  x ((x  B)  (x  A))
x ((x  A)  (x  B))  x ((x  B) v (x  A))
x ((x  A)  (x  B))  x ((x  B)  (x  A))
x ((x  A)  (x  B))  x ((x  B)  (x  A))
A
B
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Lecture 2.1 -- Sets and Set
Operations
Set Theory - Definitions and notation
Quick examples:
 {1,2,3}  {1,2,3,4,5}
 {1,2,3}  {1,2,3,4,5}
Is   {1,2,3}?
Yes! x (x  )  (x  {1,2,3})
holds, because (x  ) is false.
Is   {1,2,3}? No
Is   {,1,2,3}? Yes
Is   {,1,2,3}? Yes
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Lecture 2.1 -- Sets and Set
Operations
Set Theory - Definitions and notation
Quiz time:
Is {x}  {x}?
Is {x}  {x,{x}}?
Is {x}  {x,{x}}?
Is {x}  {x}?
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Yes
Yes
Yes
No
Lecture 2.1 -- Sets and Set
Operations
Set Theory 
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: and | are read
“such that” or
Ways to define sets “where”
Explicitly: {John, Paul, George, Ringo}
Implicitly: {1,2,3,…}, or {2,3,5,7,11,13,17,…}
Set builder: { x : x is prime }, { x | x is odd
}. In general { x : P(x) is true }, where P(x)
is some description of the set.
Ex. Let D(x,y) denote “x is divisible by y.”
Give another name for
Primes
{ x : y ((y > 1)  (y < x))  D(x,y) }.
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Lecture 2.1 -- Sets and Set
Operations
Set Theory - Cardinality
If S is finite, then the cardinality of S, |S|,
is the number of distinct elements in S.
If S = {1,2,3},
If S = {3,3,3,3,3},
If S = , |S| = 0.
|S| = 3.
|S| = 1.
If S = { , {}, {,{}} },
|S| = 3.
If S = {0,1,2,3,…}, |S| is infinite.
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Lecture 2.1 -- Sets and Set
Operations
Set Theory - Power sets
If S is a set, then the power set of S is
aka P(S)
2S = { x : x  S }.
2S = {, {a}}.
If S = {a},
If S = {a,b}, 2S = {, {a}, {b}, {a,b}}.
If S = ,
We say, “P(S) is the
set of all subsets of S.”
2S = {}.
If S = {,{}},
2S = {, {}, {{}}, {,{}}}.
Fact: if S is finite, |2S| = 2|S|. (if |S| = n, |2S|
= 2n )
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Lecture 2.1 -- Sets and Set
Operations
Set Theory - Cartesian Product
The Cartesian Product of two sets A and B is:
A x B = { <a,b> : a  A  b  B}
If A = {Charlie, Lucy, Linus}, and We’ll use these
special sets
B = {Brown, VanPelt}, then
soon!
A x B = {<Charlie, Brown>, <Lucy, Brown>,
<Linus, Brown>, <Charlie, VanPelt>, <Lucy,
VanPelt>, <Linus, VanPelt>}
a)
A1 x A2 x … x An = {<a1, a2,…, an>: a1  A1, a2 b)
c)
 A2, …, an  An}
d)
A,B finite  |AxB| = ?
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Lecture 2.1 -- Sets and Set
Operations
AxB
|A|+|B|
|A+B|
|A||B|
Set Theory - Operators
The union of two sets A and B is:
A  B = { x : x  A v x  B}
If A = {Charlie, Lucy, Linus},
and B = {Lucy, Desi}, then
A  B = {Charlie, Lucy, Linus, Desi}
B
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A
Lecture 2.1 -- Sets and Set
Operations
Set Theory - Operators
The intersection of two sets A and B is:
A  B = { x : x  A  x  B}
If A = {Charlie, Lucy, Linus},
and B = {Lucy, Desi}, then
A  B = {Lucy}
B
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A
Lecture 2.1 -- Sets and Set
Operations
Set Theory - Operators
The intersection of two sets A and B is:
A  B = { x : x  A  x  B}
If A = {x : x is a US president}, and B = {x
: x is in this room}, then
A  B = {x : x is a US president in this
room} = 
B
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A
Sets whose
intersection is
empty are called
disjoint sets
Lecture 2.1 -- Sets and Set
Operations
Set Theory - Operators
The complement of a set A is:
A = { x : x  A}
If A = {x : x is bored}, then
A = {x : x is not bored}
=
U
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
A
Lecture 2.1 -- Sets and Set
Operations
= U
and
U=
Set Theory - Operators
The set difference, A - B, is:
U
B
A
A-B={x:xAxB}
A-B=AB
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Lecture 2.1 -- Sets and Set
Operations
Today’s Reading
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Rosen 2.1
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Lecture 2.1 -- Sets and Set
Operations