Lecture 2.1: Sets and Set Operations CS 250, Discrete Structures, Fall 2014 Nitesh Saxena Adopted from previous lectures by Cinda Heeren

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


Operations

Outline

Set Definitions and Theory Set Operations


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: {}


Operations


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


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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


Operations


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


Operations


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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Quiz time:

Is {x} {x}?

Is {x} {x,{x}}?

Is {x} {x,{x}}?

Is {x} {x}?

Yes

Yes

Yes

No


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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


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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.


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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


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


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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


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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


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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


Operations


The set difference, A - B, is:

AU

B

A - B = { x : x A x B }

A - B = A B


Operations

Today’s Reading Rosen 2.1

Documents

Lecture 2.1: Sets and Set Operations CS 250, Discrete Structures, Fall 2014 Nitesh Saxena Adopted from previous lectures by Cinda Heeren