Lecture 2.1: Sets and Set Operations CS 250, Discrete Structures, Fall 2014 Nitesh Saxena Adopted from previous lectures by Cinda Heeren
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Lecture 2.1: Sets and Set Operations
CS 250, Discrete Structures, Fall 2014
Nitesh Saxena
Adopted from previous lectures by Cinda Heeren
04/21/23Lecture 2.1 -- Sets and Set
Operations
Course Admin HW1 Due
11am 09/18/14 – this Thursday Please follow all instructions Recall: late submissions will not be
accepted
04/21/23Lecture 2.1 -- Sets and Set
Operations
Outline
Set Definitions and Theory Set Operations
04/21/23Lecture 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).
= {} is the empty set or null set, or the set containing no elements.
U: is the set of all possible elements in the universe
Note: {}
04/21/23Lecture 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.”
Venn Diagram
or, “B contains A.”or, “every element of A is also in
B.”or, x ((x A) (x B)).
A
B
04/21/23Lecture 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 Aiff, 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
04/21/23Lecture 2.1 -- Sets and Set
Operations
Set Theory - Definitions and notation
A B means “A is a proper subset of B.” 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
04/21/23Lecture 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
04/21/23Lecture 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}?
Yes
Yes
Yes
No
04/21/23Lecture 2.1 -- Sets and Set
Operations
Set Theory - Ways to define sets
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
{ x : y ((y > 1) (y < x)) D(x,y) }.
: and | are read “such that” or
“where”
Primes
04/21/23Lecture 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}, |S| = 3.
If S = {3,3,3,3,3},
If S = ,
If S = { , {}, {,{}} },
|S| = 1.
|S| = 0.
|S| = 3.
If S = {0,1,2,3,…}, |S| is infinite.
04/21/23Lecture 2.1 -- Sets and Set
Operations
Set Theory - Power sets
If S is a set, then the power set of S is 2S = { x : x S }.
If S = {a},
aka P(S)
If S = {a,b},
If S = ,
If S = {,{}},
We say, “P(S) is the set of all subsets of S.”
2S = {, {a}}.
2S = {, {a}, {b}, {a,b}}.2S = {}.
2S = {, {}, {{}}, {,{}}}.
Fact: if S is finite, |2S| = 2|S|. (if |S| = n, |2S| = 2n)
04/21/23
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 B = {Brown, VanPelt}, then
A,B finite |AxB| = ?
A1 x A2 x … x An = {<a1, a2,…, an>: a1 A1, a2 A2, …, an An}
A x B = {<Charlie, Brown>, <Lucy, Brown>, <Linus, Brown>, <Charlie, VanPelt>, <Lucy, VanPelt>, <Linus, VanPelt>}
We’ll use these special
sets soon!
a) AxBb) |A|+|B|c) |A+B|d) |A||B|
Lecture 2.1 -- Sets and Set Operations
04/21/23Lecture 2.1 -- Sets and Set
Operations
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}, thenA B = {Charlie, Lucy, Linus, Desi}
AB
04/21/23Lecture 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}, thenA B = {Lucy}
AB
04/21/23Lecture 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} =
ABSets whose
intersection is empty are called
disjoint sets
04/21/23Lecture 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}
A
=
= U and U =
U
04/21/23Lecture 2.1 -- Sets and Set
Operations
Set Theory - Operators
The set difference, A - B, is:
AU
B
A - B = { x : x A x B }
A - B = A B
04/21/23Lecture 2.1 -- Sets and Set
Operations
Today’s Reading Rosen 2.1
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