Sets and Functionscourses.cse.tamu.edu/davidkebo/csce-222/lectures/CSCE222... · 2019-09-17 · •Relations:sets of ordered pairs represent relationships between objects •Graphs:sets

CSCE 222Discrete Structures for Computing

David Kebo Houngninou

Sets and Functions

Chapter 2

Outline

• Sets• Sets operations• Relations• Functions

CSCE 222 Discrete Structures for Computing 3Houngninou

Sets

What is a set?

Houngninou CSCE 222 Discrete Structures for Computing 4

Applications

• Combinations: unordered collections of objects• Relations: sets of ordered pairs represent relationships

between objects• Graphs: sets of ordered or unordered pairs also represent

relationships • Functions


The Language of Sets

Houngninou CSCE 222 Discrete Structures for Computing

Let S be the set of all integers between 1 and 5, inclusive S = {1,2,3,4,5}We want to determine if x is an element of set S: x Î S?

Let x = 3; is x Î S? YesNow let x = 10; is x Î S? No, so we denote this as x Ï S

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The Language of Sets


Let set T = {5, 4, 3, 2, 1} and S = {1, 2, 3, 4, 5}is set T = set S?

Do sets T and S contain the same elements?

Does {2} = 2?{2} is a set with the element 2, while 2 is the number 2

7

Common Sets


C = set of complex numbersR = set of real numbersR+ = set of positive real numbers

Q = set of rational numbersZ = integers = {…,-3,-2,-1,0,1,2,3,…}Z⁺ = positive integers = {1,2,3,…}N = natural numbers = {0,1,2,3….}

8

Set-Roster Notation


Set S = {1,2,3,4,5} For a small set, we can write all the elements in the set. This is the set-roster notation (roster of elements).

For a large set, we can write A = {1,2,3,….100}.That is all integers from 1 to 100, inclusive.

For some sets (e.g. real numbers), the roster notation is impractical.

10

Set-Builder Notation


We define the set of all elements x in set S such that P(x) is true as: {x Î S | P(x)}1. x Î S means “elements x in set S”2. | means “such that”3. P(x) means that property P(x) is satisfied (true)

e.g. {x Î Z | 2 < x ≤ 5} = {3, 4, 5}

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Set Equality


When are two sets equal?

12

Set Equality


B = {1,2,3,4,5}D = {1,2,3,4,5}E = {1,2,3,4}

Is B equal to D?

Is B equal to E?

Set Equality – Formal Definition

Given sets A and B, A equals B (A = B) if and only if:Every element of A is in B, andEvery element of B is in A

∀x [x ∈ A ⟷ x ∈ B]


( ) ( )ABBABA ÍÙÍÛ=

The Empty Set { } or ⌀


⌀ or { } is the empty set, the set that has no elements.The empty set has a cardinality of 0.

Note: a set with one element is called the singleton set.

Do not get confused! ⌀ or { } means the empty set.{⌀} is a singleton set containing the empty set.

15

Venn diagrams are visual representations of sets.The Universal set (all objects under consideration) is a rectangle. Particular sets are circles inside the rectangle.

Venn Diagrams


U

Set A is a subset of set B if every element of A is in B.Notation: A ⊆ B

Subsets


U

A

B

Subsets


Assume we have the following sets:A = Z (set of integers)B = {1,2,3,4,5} (set-roster notation)C = {x Î Z | 2 < x < 5} (set-builder notation)

All the elements of set B are in set A; B is a subset of A: B Í A

B Í A means that for all x, if x Î B, then x Î A

What about set C ?

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Set A is a proper subset of B if A is a subset of B but is not equal to B. Notation: A ⊂ BEvery element of A is in B and there is at least one element of B that is not in A.

Proper subsets


U

A

B

Subset vs. Proper Subset


Assume we have the following sets:B = {1,2,3,4,5}D = {1,2,3,4,5}E = {1,2,3,4}Note that D = B

Recall the definition of a subset: ∀x, if x Î D, then x Î B.We can state that D is a subset of B.We can also state that B is a subset of itself.However, E is a proper subset of B since there is at least one element in B that is not in E (5).

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Power set


The power set of a set S is the set of all subsets of S.Notation: P(S)e.g. • The power set of ⌀ (cardinality 0) is {⌀} (cardinality 1)• The power set of {0} (cardinality 1) is {⌀, {0}} (cardinality 2)• The power set of {0,1} (cardinality 2) is {⌀,{0},{1},{0,1}}

(cardinality 4)

Note: if |S| = n, then |P(S)| = 2n

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Operations on Sets

Earlier, we discussed operations on statement variables (∨, ∧,∼)

There are similar operations that we perform on sets:• Union• Intersection• Complement• Difference


Sets Union

• The union of sets A and B is the set of all elements that are in at least one of A or B

• Denoted as A ∪B


Shaded regionrepresents A È B.

Sets Intersection

• The intersection of sets A and B is the set of all elements that common to both A or B

• Denoted as A ∩B


Shaded regionrepresents A Ç B.

Sets Difference

• The difference B minus A is the set of all elements that are in B and not A

• Denoted as B - A


Shaded regionrepresents B – A.

Set Complement

• The complement of A is the set of all elements in U that are not in A

• Denoted as Ac


Shaded regionrepresents Ac.

Formal Definitions of Set Operators


{ }{ }{ }{ }AxUxA

AxBxUxABBxAxUxBABxAxUxBA

c ÏÎ=

ÏÙÎÎ=-ÎÙÎÎ=ÇÎÚÎÎ=È

||||

Set identities are similar to the logical equivalences we learned in Ch.1. Here are De Morgan’s Laws for sets:

We also have identity, domination, idempotent, double complementation, commutative, associative, absorption, complement.Read Table 1 in Section 2.2, page 130

Set Identities


Relations

Cartesian Product


The Cartesian product of sets A and B is the set of all ordered pairs (a, b) where a is in set A and b is in set B

Or:

Given A = {3,5,7}, B = {2,4}A⨯B = {(3,2), (3,4), (5,2), (5,4), (7,2), (7,4)}B⨯A = {(2,3), (2,5), (2,7), (4,3), (4,5), (4,7)}B⨯B = {(2,2), (2,4), (4,2), (4,4)}

If |A| = n and |B| = m, What is |A × B|?

( ){ }BbAabaBA ÎÎ=´ and |,

31

Relations


Given two sets A and B: A relation R from A to B is a subset of A⨯B.

R Í A⨯B

Given an ordered pair (x,y) in A⨯B:x is related to y by R iff (x,y) is in R

A is the domain of R and B is the co-domain of R

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Arrow Diagram of a Relation


We can also represent relations graphically using an arrow diagram:

Represent sets as regions, with elements of sets as points in region.

For each x in the domain region and y in the co-domain region, draw an arrow from x to y iff (x,y) Î R

34

Let A = {1, 2, 3} and B = {1, 3, 5} For all (x, y ) ∈ A ´ BLet’s define relations S and T from A to B as follows:

Draw arrow diagrams for S and T.

Example: Arrow Diagrams of Relations


Example: Solution

It is possible to have arrows from the same element of A pointing to different directions.It is possible to have an element of A that does not have an arrow coming out of it.


Functions

Functions


A function f from set A to set B is an assignment of exactly oneelement of B to each element of A. f: A → B and f(a) = b.

Ways to specify functions:• give a formula, e.g. f(x) = x + 1• give an algorithm to compute the function

Every function is a relation but not vice versa!

38

Function Important Terminologies

For a function f: A→B with f(a) = b

• A is the domain• B is the codomain• b is the image of a • a is a preimage of b (preimage is not necessarily unique) • B is the range of f (images of all elements of A)


Function Terminologies

Suppose f: Z → Z is f(x) = x2

• Domain is Z (set of integers)• Codomain is Z• Image of 3 is 9• Preimages of 16 are 4 and -4• Range is the set of all integers that are perfect squares


Arrow Diagrams

An arrow diagram defines a function if the diagram has the following properties:

1. Every element of X has an arrow coming out of it.2. No element of X has two arrows to two different

elements of Y.


Example


X = {a, b, c}, Y = {1, 2, 3, 4}We define a function f from X to Y as follow:


What is the domain and co-domain of f?

The domain X of f is {a, b, c}The co-domain Y of f is {1, 2, 3, 4}


What is the range of f ?

The range of f is defined as {y ∈ Y | y = f(x), for x in X}i.e., what values of Y are mapped to by X?The range is {2,4}


Is c a preimage of 2? Yes, since f(c) = 2

Is b a preimage of 3?No, since f(b) ≠ 3

Function Equality

• Given functions F: X→Y and G: X→Y• Then:


XxxGxFGF Î"=Û= )()(

Classes of functions

• One-to-One Functions• Onto Functions• One-to-One Correspondence• Inverse Functions


One-to-One Functions


Suppose we have the following two functions

(1)

(2)



For function (1), each x maps to the same y: F(x1) = F(x2)

(1)



(2)

For function (2), each x maps to a unique y: F(x1) ≠ F(x2)

Function (2) is called a one-to-one function.

One-to-One Functions: Definition

• Let F be a function from set X to set Y• F is one-to-one iff for all elements x1 and x2 in X:

– If F(x1) = F(x2), then x1 = x2, or– If x1 ≠ x2, then F(x1) ≠ F(x2)

• Symbolically:

Also called an “injective function” or an “injection”


( ) ( )( ) ( )212121 ,,one-to-one is : xxxFxFXxxYXF =®=Î"Û®

Onto Functions


Suppose we have the following two functions

(1)

(2)

Onto Functions


For function (1), there is at least one y that is not F(x) for any x

(1)

Onto Functions


For function (2), every y = F(x) for at least one xFunction (2) is called an onto function.

(2)

Onto Functions Definition

• Let F be a function from set X to set Y• F is onto iff

– Given any y in Y, there is at least one x in X where y = F(x)• Symbolically:

Also called a surjective function or a surjection


yxFXxYyYXF =Î$Î"Û® )(|,onto is :

One-to-One Correspondence


Suppose we have the following function

Is F one-to-one? Yes, each F(x) has a distinct x Is F onto? Yes, for all y, there is an x such that y = F(x)Therefore, there is a one-to-one correspondence from X to Y

One-to-One Correspondence• Let F be a function from set X to set Y

• F has one-to-one correspondence from set X to set Y iff F is one-to-one, and F is onto

Also called a bijection.


Inverse Function

• Suppose function F has a one-to-one correspondence from set X to set Y

• Then there is a function from Y to X that “undoes” the action of F

• This function is called the inverse function for F


Inverse Function Definition

Given F: X→Y that is a one-to-one correspondence

There is a function F-1: Y→X where

F-1(y) is called the inverse function for F


)()(1 xFyxyF =Û=-

Inverse Function Arrow Diagram


Inverse Function Example

F(x) = 4x – 5

1. Each x maps to a unique y, so F is one-to-one

2. Every y = F(x) for some x, so F is onto

Thus, F has a one-to-one correspondence

Therefore, we can determine F-1


Inverse Function Example


45)(

4554)(

1 +==

=+-==

- yxyF

xyxyxF

Floor and Ceiling

Floor and ceiling functions work on real numbers For a real number x:• The floor of x rounds x down to the nearest integer value• The ceiling of x rounds x up to the nearest integer value



x = 3.141592654

Ceiling(x) = 4

Floor(x) = 3

Round UP to nearest integer

Round DOWN to nearest integer

Floor Definition


Ceiling Definition



x = - 3.141592654

Ceiling(x) = -3

Floor(x) = -4

Round UP to nearest integer

Round DOWN to nearest integer

Examples


1. How many bytes are needed to store 100 bits of data? ⌈100/8⌉ = ⌈12.5⌉ = 13 bytes

2. A cell is a unit of 53 bytes. If the data rate is 500 kilobits per second, how many complete cells can be transmitted in 1 minute?

In 1 minute, 500⨯1000⨯60 = 30Mbits can be transmitted.Each cell is 53⨯8 = 424bits.⌊30M/424⌋ = 70,754 complete cells

Application to C/C++/Java


double x, x1, x2;

x1 = floor(x); // floor of xx2 = ceil(x); // ceiling of x

End


Documents

Sets and Functionscourses.cse.tamu.edu/davidkebo/csce-222/lectures/CSCE222... · 2019-09-17 · •Relations:sets of ordered pairs represent relationships between objects •Graphs:sets