Set Theory - Section 2

Solution l(4) = 1/4. Therefore, l-1(1/4) = 4.

Let X = {1, 4, 5, 7}. Define l: X → A to be l(x) = 1/x. What is l-1(1/4)?

m(0101) = 1010, so m-1(1010) = 0101.

Let m: {0,1}4 → {0,1}4. For x ∈ {0,1}4, m(x) is obtained by reversing the order of the bits. What is m-1(1010)?

Yes Natural numbers are a proper subset of integers. Integers are a proper subset of rational numbers. Rational numbers are a proper subset of real numbers.

N ⊂ Z ⊂ Q ⊂ R ? Yes No

Is A ⊆ p(A) for any A?

No , notice our pic not every set has A in it

see pic

Properties of exponents

pic

Properties of logarithms

1. List the elements of D given the below given Z+ means positive integers D = {x ∈ Z+ | x < 6} 2. What is the cardinality of D? 3. What is the cardinality of {∅,{a,b}}?

1. 1,2,3,4,5 2. | D | = 5 3. 2, look at the commas, there are 2 sets within this set.

T or F for each: {a,b,a,a} ⊆ {a,b,c} {c,d} ⊆ {c,d} {a} ⊆ {{a}} ∅ ⊆ {x,y,z}

1. True, remember we can reduce repeats 2. True 3. False 4. True, empty set is a subset of every set

p(∅) = ?

1. p(∅) = {∅} or {{}} so 1 |∅| = 0 |p(∅)| = 2^0 = 1

Elements and Cardinality Let C = {yellow, blue, red} How do we write? 1. Yellow is an element of C? 2. Green is not an element of C? 3. The cardinality of C is 3?

1. yellow ∈ C 2. green ∉ C 3. |C| = 3

p({∅}) = ?

2. |{∅}| = 1 |p({∅})| = 2^1 = 2

Set Builder Notation Even integers 2Z = {..., -4, -2, 0, 2, 4,...} better way to express this?

2Z = {..., -4, -2, 0, 2, 4,...} becomes>>> ={2n | n ∈ Z}

If |A| = m, what is |p(p(p(A)))|?

2^2^2^m

Let |c| = k and |d| = j, what is |p(cxd)| ?

2^|cxd| or 2^kj

The cartesian product A x B, is the set: {(a,b) | a ∈ A and b ∈ B} Given x = {0,1,2} and y = {0,1} What is the cardinality of A x B?

6

bit, n-bit string.

A ___ is a character in a binary string. A string of length n is also called an _____.

binary

A _____ string is a string whose alphabet is {0, 1}.

partition

A ______ of a non-empty set A is a collection of non-empty subsets of A such that each element of A is in exactly one of the subsets

surjective

A function f: X → A is ______ or onto if the range of f is equal to the target A. That is, for every a ∈ A, there is an x ∈ X such that f(x) = a.

injective

A function f: X → A is ______or one-to-one if x1 ≠ x2 implies that f(x1) ≠ f(x2). That is, f maps different elements in X to different elements in A.

invertible

A function f: X → Y is ______ if there exists a function g with domain Y and range X with the property f(x) = y ⇔ g(y) = x.

bijective

A function is ______ if it is both injective and surjective. A bijective function is called a bijection. A bijection is also called a one-to-one correspondence.

The intersection of sets A and B is written as ...

A ∩ B

The union of sets A and B is written as...

A ∪ B

Given sets A and B the difference A-B, is deined as ...

A-B ={x| x ∈ A and x ∉ B}

Well-Ordering Principle

Any non empty subset of N has a least element. If we have Z however it does not work because Z goes to negative infinity.

Not equal f's target set is R, while g's target set is R+. If f = g, then f and g must have the same target.

Are these functions equal? f: R → R, f(x) = x2 + 1 g: R → R+, g(x) = (x + 1)2 - 2x

A ⊕ B ⊕ C = {1, 5, 6, 7}. Note that the symmetric difference operation is associative: (A ⊕ B) ⊕ C = A ⊕ (B ⊕ C). Therefore since 1 is in all 3 it is included because its not in just one or the other

Calculate A ⊕ B ⊕ C for A = {1, 2, 3, 5}, B = {1, 2, 4, 6}, C = {1, 3, 4, 7}

To check that f is injective, check that x≠x′, implies 3x−2≠3x′−2. Equivalently, show the contrapositive: 3x−2=3x′−2, implies x=x′. f is injective because if 3x−2=3x′−2, then x=x′. f is surjective because for every y, there is an x such that 3x−2=y. The inverse of f can be found by solving for an expression that gives the value of x in terms of y. The inverse of f on input y is (y+2)/3.

Check if the function is bijective and then give us the inverse: f: R → R f(x) = 3x - 2

Let f be a function that assigns employees to offices in a company. Let g be the function that maps each office to the telephone number for the phone in that office. An employee, Rajiv, is assigned the office f(Rajiv). Rajiv's work phone number is g(f(Rajiv)). The process of applying a function to the result of another function is called composition.

Composition of functions

T. 3 ∈ A, so { 3 } ⊆ A. F. { 1, 2, 3 } ∈ A. However, 1 ∉ A, so { 1, 2, 3 } ⊈ A.

Consider this set A = { 3, 4, { 3, 4 }, { 1, 2, 3 }, 5 } { 3 } ⊆ A. T or F? { 1, 2, 3 } ⊆ A T or F?

1. 13 2. 64 3. 4 4. 17

Define functions f, g, h, all of which have R as their domain and R as their target. f(x) = 3x + 1 g(x) = x2 h(x) = 2x 1) What is (f ο g)(2)? 2) What is (g ο h)(3)? 3) What is (f ο g ο h)(0)? 4) What is (f ο f-1)(17)?

(a, a, a, a, a) is a 5-tuple and A4 contains only 4-tuples.

Define the set A = { a, b, c } T or F? (a, a, a, a, a) ∈ A4

0011 t = 001, so putting the string t together with the bit 1 results in 0011.

Define the string t = 001. What is t1?

001 Concatening λ to a string does not add or remove any bits to the string, so tλ = t.

Define the string t = 001. What is tλ?

The complement of A is ...

Everything outside A but within the universe see pic for how complement is written A' is one example

Remember {0, 1}0 = {λ} so we include an extra 0 in all. { 000, 001, 010, 011 }

Express the following sets using the roster method. {0x: x ∈ {0, 1}^2}

T or F? Sets have an order to them.

F.

The floor function and the ceiling function map real numbers onto integers. The floor and ceiling functions round real numbers to a nearby integer in different ways.

Floor and Ceiling Functions

range

For function f: X → Y, an element y is in the ____ of f if and only if there is an x ∈ X such that (x, y) ∈ f

Many functions are mathematical functions that map numbers to numbers, such as the function x2, which maps a number to its square.

Function A function maps elements from one set X to elements of another set Y.

inverse, the function must be bijective for this to work

If a function f: X → Y is a bijection, then the______ of f is obtained by exchanging the first and second entries in each pair in f. The inverse of f is denoted by f^-1.

Yes

If one value of Y does not come from X but X is connected to all other Y values once only is F a function?

If |A| = n, then |p(A)| =

If |A| = n, then |p(A)| = 2^|A| or 2 ^n 2 because there a 2 options, add the element or not

Special cases of functions are those that relate the domain to the range in such a way that every element in the domain is mapped to a different element in the range or that every element in the range is mapped from an element in the domain. These functions are called injective and surjective functions, respectively.

Injective, surjective, and bijective functions

w

Let X = { u, v, w, x }. Define a function g: X → X to be: g = { (u, v), (v, w), (w, x), (x, u) }. What is g-1(x)?

2^6 = 64, so log2 64 = 6.

what is the value of log2 64?

Set Builder Notation Q = {..., 1/1, 1/2, 1/3, 2/3, etc.} better way to express this?

Q = {..., 1/1, 1/2, 1/3, 2/3, etc.} becomes>>> ={m/n | m,n ∈ Z, n ≠ 0} define elements as variables so this set has elements of form of m/, such that (the "|" means such that ) m and n have to be integers (m,n ∈ Z) and n cannot be zero.

B.

See pic for choices The universal set is Z+. A1 = { x ∈ Z+: x is prime } A2 = { x ∈ Z+: x is odd and x is a composite number } A3 = { x ∈ Z+: x is even } A4 = { 1 }

The empty set ∅ is not the same as { ∅ }. The cardinality of { ∅ } is one since it contains exactly one element, which is the empty set. A set can contain a combination of numbers and sets of numbers as in: B = { 2 , ∅, { 1, 2, 3 }, { 1 } } Then 2 ∈ B, but 1 ∉ B. Also, { 2 } ⊆ B, but { 1 } ⊈ B.

Set of sets It is possible that the elements of a set are themselves sets. For example, consider the set A: A = { { 1, 2 }, ∅, { 1, 2, 3 }, { 1 } } The set A has four elements: { 1, 2 }, ∅, { 1, 2, 3 }, and { 1 }. For example, { 1, 2 } ∈ A. Note that 1 is not an element of A, so 1 ∉ A, although { 1 } ∈ A. Furthermore, { 1 } ⊈ A since 1 ∉ A.

T or F? In a set repeated elements are listed once.

T

empty

The ____ string is the unique string whose length is 0 and is usually denoted by the symbol λ. Since {0, 1}0 is the set of all binary strings of length 0, {0, 1}0 = {λ}.

identity function

The _____ function always maps a set onto itself and maps every element onto itself.

see pic

The exponential function is one-to-one and onto, and therefore has an inverse. The logarithm function is the inverse of the exponential function.

The set of natural numbers: All integers greater than or equal to 0.

The set N is...

The set of rational numbers: All real numbers that can be expressed as a/b, where a and b are integers and b ≠ 0. 0, 1/2, 5.23, -5/3

The set Q is...

The set of real numbers. 0, 1/2, 5.23, -5/3, π, etc. Real Numbers include: Whole Numbers (like 0, 1, 2, 3, 4, etc) Rational Numbers (like 3/4, 0.125, 0.333..., 1.1, etc ) Irrational Numbers (like π, √2, etc ) Real Numbers can also be positive, negative or zero.

The set R is...

The set of all integers...., -2, -1, 0, 1, 2, ...

The set Z is...

The empty set ∅ = {} | ∅ | = 0 |{∅}| = 1 This means what is the size of a set containing this set

The size is zero but if a set contains this empty set its size is 1

see pic

The symmetric difference between two sets, A and B, denoted A ⊕ B, is the set of elements that are a member of exactly one of A and B, but not both.

equal

Two functions, f and g, are ____ if f and g have the same domain and target, and f(x) = g(x) for every element x in the domain. The notation f = g is used to denote the fact that functions f and g are equal

disjoint, pairwise disjoint

Two sets, A and B, are said to be _____ if their intersection is empty (A ∩ B = ∅). A sequence of sets, A1, A2, ..., An, is ______ ______ if every pair of distinct sets in the sequence is disjoint (i.e., Ai ∩ Aj = ∅ for any i and j in the range from 1 through n where i ≠ j).

Set Operations and Venn Diagrams Every Set A exists within some universe U

U could be something like Z, R, Z+

see pic

Well defined Functions If one value of X maps to two different values of y, x is no longer a function

When talking about Cartesian Products we usually talk about cross products which are a set. The cartesian product A x B, is the set: {(a,b) | a ∈ A and b ∈ B} Given x = {0,1,2} and y = {0,1} find X x Y =

X x Y = {(0,0),(0,1)(1,0)(1,1)(2,0)(2,1)}

Is A ∈ p(A) for any A?

Yes A is an element of power set A notice in the pic A = {a,b}

Composite function

f and g are two functions, where f: X → Y and g: Y → Z. The composition of g with f, denoted g ο f, is the function (g ο f): X → Z, such that for all x ∈ X, (g ο f)(x) = g(f(x)).

x

f(f^-1(x)) = ?

The result of solving the equation y = x/(x + 1) for x is x = -y/(y-1). Therefore, f-1(y) = -y/(y-1) which is the same as f-1(x) = -x/(x-1).

f(x) = x/(x + 1), x ≠ -1, with a range of f(x) ≠ 1. What is f-1?

The result of solving the equation y = -x + 3 for x is x = -y + 3. Therefore, f-1(y) = -y + 3 which is the same as f-1(x) = -x + 3.

f: R → R, where f(x) = -x + 3. What is f^-1?

1.(x + 2)^3 2. x^3+2

f: R+ → R+, f(x) = x^3 g: R+ → R+, g(x) = x + 2 Then 1. (f ο g)(x) = f(g(x)) =? 2. (g ο f)(x) = g(f(x)) = ?

The result of solving the equation y = 2x2 - 4 for x is x = ((y+4)/2)1/2. Therefore, f-1(y) = ((y+4)/2)1/2 which is the same as f-1(x) = ((x+4)/2)1/2.

f: R+ → R+, where f(x) = 2x2 - 4. What is f^-1?

domain, target

f: X → Y is the notation to express the fact that f is a function from X to Y. The set X is called the _____ of f, and the set Y is the _____ of f. The fact that f maps x to y (or (x, y) ∈ f) can also be denoted as f(x) = y.

f-1(x)= x^1/3

f:R → R, where f(x) = x3. What is f^-1?

list the elements of p(p(∅))

first do p(∅) which is {∅} then do p({∅}) which is the empty and set the set containing the empty set {∅,{∅}}

Both Correct If x ≠ y, thenx - 4 ≠ y - 4(injective). For every y, there is an integer x such thatx - 4 = y(onto).

h: Z → Z. h(x) = x - 4. see pic for options

1. 5 2. 6 3 -5

see pic

A power set of A, p(A), is the set containing all possible subsets of A. A = {a,b} p(A) = {∅,{a},{b}{a,b}}

see pic

Cartesian Products An ordered pair (a,b) is a set {{a},{a,b}} youve seen ordered pairs before as graph coordinates (1,2) = {{1},{1,2}} (2,1) = {{2},{2,1}} (-2,0) = etc.

see pic

Cartesian products can generalize to n-tuples 3-tuple would be A x B x C

see pic

Exercise A = {1,3,5,7,9} B = {4,8,12,16} C = {1,4,9,16} A ∪ B = C ∩ B = C-B = ∅ ∩ B =

see pic

Exercise: Let A ={a,b} and B ={c,d} AxB = ? B^2 = ? ∅ x A = ?

see pic

If |B| = m and |A| = n then find... |AxB| = ? |A^2| = ? |B^32xA^19| = ?

see pic

Indexed Sets How can we shorten A1∩(A2∩(A3∩(A4∩A5)))? What about for unions?

see pic

Subsets If A is a subset of B, then every element in A must also be in B. A⊆B

see pic. A and B can also be exactly the same size if the notation is written A⊆B However A c B (no line under the "c") means A must be smaller

A ___ is a collection of elements. They can be finite or infinite.

set

Injective only Correct If x ≠ y, then 2x ≠ 2y (injective). For any odd integer y, there is no x such that2x = y.For example, there is no integer x such that2x = 5(not onto).

t: Z → Z. t(x) = 2x.