Set

 

Set comprehension

Set comprehension is a concise way to define a set in mathematical notation. It is a way to describe the elements of a set in terms of a property that they all share.

For example, the following set comprehension defines the set of all even numbers:

{x | x is an even number}

This set comprehension can be read as "the set of all x such that x is an even number".

Another example of a set comprehension is the following:

{x^2 | x is a real number}

This set comprehension defines the set of all squares of real numbers.

Subsets

A subset of a set S is a collection of elements of S that may or may not include all of the elements of S. For example, the set {1, 2, 3} is a subset of the set {1, 2, 3, 4, 5}.

To construct subsets of a given set, we can use the following methods:

• The empty set: The empty set is a subset of every set.
• The power set: The power set of a set S is the collection of all subsets of S.
• Set operations: We can also construct subsets of a given set using set operations such as union, intersection, set difference, and complement.

Closed and open intervals

A closed interval is an interval that includes its endpoints. For example, the closed interval [1, 3] includes the numbers 1, 2, and 3.

An open interval is an interval that does not include its endpoints. For example, the open interval (1, 3) does not include the numbers 1 or 3.

Set operations

The four basic set operations are union, intersection, set difference, and complement.

• Union: The union of two sets A and B is the collection of all elements that are in either A or B or both.
• Intersection: The intersection of two sets A and B is the collection of all elements that are in both A and B.
• Set difference: The set difference of two sets A and B is the collection of all elements that are in A but not in B.
• Complement: The complement of a set A is the collection of all elements that are not in A.

Notation:

• The union of two sets A and B is denoted by A ∪ B.
• The intersection of two sets A and B is denoted by A ∩ B.
• The set difference of two sets A and B is denoted by A \ B.
• The complement of a set A is denoted by A'.

Examples:

• Let A = {1, 2, 3} and B = {4, 5, 6}.
• Then, A ∪ B = {1, 2, 3, 4, 5, 6}.
• A ∩ B = {}.
• A \ B = {1, 2, 3}.
• B \ A = {4, 5, 6}.
• A' = {4, 5, 6}.

I hope this helps!

Membership of sets

The membership of a set is the collection of all elements that are in the set. An element is a member of a set if it is present in the set.

To determine whether an element is a member of a set, we can use the following notation:

x ∈ A

where x is the element and A is the set. This notation means "x is an element of A".

For example, the following statement is true:

2 ∈ {1, 2, 3}

This statement is true because the number 2 is present in the set {1, 2, 3}.

Set comprehension

Set comprehension is a concise way to define a set in mathematical notation. It is a way to describe the elements of a set in terms of a property that they all share.

For example, the following set comprehension defines the set of all even numbers:

{x | x is an even number}

This set comprehension can be read as "the set of all x such that x is an even number".

Another example of a set comprehension is the following:

{x^2 | x is a real number}

This set comprehension defines the set of all squares of real numbers.

Set comprehension combines generators, filters, and transformations to produce new sets from old sets.

• Generators: Generators produce a sequence of elements. For example, the generator range(10) produces the sequence of elements [0, 1, 2, ..., 9].
• Filters: Filters select elements from a sequence based on a condition. For example, the filter lambda x: x % 2 == 0 selects even numbers from a sequence.
• Transformations: Transformations apply a function to each element of a sequence. For example, the transformation lambda x: x^2 squares each element of a sequence.

To construct a set using set comprehension, we use the following syntax:

{generator | filter | transformation}

For example, the following set comprehension defines the set of all squares of even numbers less than 10:

{x^2 | x in range(10) | x % 2 == 0}

This set comprehension can be read as "the set of all x such that x is a square of an even number less than 10".

I hope this helps! 1. Compare the cardinalities of two infinite sets.                                                                                                              2. Know what countable sets are.

3. Explain why the cardinalities of $\mathbb{�},\mathbb{�}$, and $\mathbb{�}$ are the same.

4. Argue that the real numbers are not countable.

Expressing natural numbers as sets

One way to express natural numbers as sets is to use the following definition:

0 = {}
1 = {0}
2 = {1}
3 = {2}
...

In other words, the natural number 1 is the set containing the empty set, the natural number 2 is the set containing the set containing the empty set, and so on.

This definition is called the von Neumann definition of natural numbers. It is a recursive definition, which means that it defines each natural number in terms of the smaller natural numbers.

Russell's paradox

Russell's paradox is a paradox that shows that not every collection can be called a set.

The paradox goes as follows:

1. Define the set R as the collection of all sets that are not members of themselves.
2. Is R a member of itself?
   • If R is a member of itself, then it is not a member of itself, because it is a member of the collection of all sets that are not members of themselves.
   • If R is not a member of itself, then it is a member of itself, because it is a member of the collection of all sets that are not members of themselves.

Either way, we have a contradiction. Therefore, the set R cannot exist.

This paradox shows that there cannot be a set of all sets.

Class

A class is a collection of objects that is not a set.

Classes can be infinite, but they can also be proper classes, which are classes that are too large to be sets.

For example, the class of all sets is a proper class. This is because the set of all sets cannot be a member of itself, as shown by Russell's paradox.

Another example of a proper class is the class of all natural numbers. This is because the natural numbers cannot be collected into a set, as shown by the Burali-Forti paradox.

Classes are important in mathematics, but they are more difficult to work with than sets. This is because classes do not have the same properties as sets. For example, the union and intersection of two classes are not necessarily classes.

I hope this helps!

Comparing the cardinalities of two infinite sets

To compare the cardinalities of two infinite sets, we use the concept of a bijection. A bijection between two sets A and B is a one-to-one and onto mapping between the two sets.

If there is a bijection between two sets A and B, then the two sets have the same cardinality.

If there is no bijection between two sets A and B, then the two sets have different cardinalities.

Countable sets

A countable set is a set that has the same cardinality as the set of natural numbers.

Examples of countable sets include:

• The set of natural numbers: {1, 2, 3, 4, ...}
• The set of integers: {..., -3, -2, -1, 0, 1, 2, 3, ...}
• The set of rational numbers: {p/q | p and q are integers, q ≠ 0}

Cardinalities of N, Z, and Q

The cardinalities of N, Z, and Q are the same. This is because there is a bijection between each of these sets.

For example, the following function is a bijection between N and Z:

f(n) = if n is even, then n/2 else -(n + 1)/2

This function maps each natural number to a unique integer.

The following function is a bijection between Z and Q:

f(n) = n/1

This function maps each integer to a unique rational number.

Real numbers are not countable

The real numbers are not countable. This can be proven by using the Cantor diagonalization argument.

The Cantor diagonalization argument shows that there cannot be a bijection between the set of natural numbers and the set of real numbers.

Therefore, the cardinality of the real numbers is greater than the cardinality of the natural numbers.

I hope this helps!

Number of elements in the Cartesian product of two finite sets

The Cartesian product of two finite sets A and B is the set of all ordered pairs (a, b) where a is an element of A and b is an element of B.

The number of elements in the Cartesian product of two finite sets A and B is equal to the product of the number of elements in A and the number of elements in B.

For example, if A = {1, 2, 3} and B = {4, 5}, then the Cartesian product of A and B is the set {(1, 4), (1, 5), (2, 4), (2, 5), (3, 4), (3, 5)}. This set has 6 elements, which is equal to the product of the number of elements in A (3) and the number of elements in B (2).

Relations as a subset of the Cartesian product

Definition of a function, domain, codomain, and range of a function:

• A function is a relation between two sets A and B, where each element of A is related to exactly one element of B.
• The domain of a function is the set of all input values for the function.
• The codomain of a function is the set of all possible output values for the function.
• The range of a function is the set of all output values that the function actually produces.

Identifying the domain and range of a given function:

To identify the domain and range of a given function, we can use the following steps:

1. Identify any restrictions on the input values of the function. For example, if the function contains a square root radical, then the input value cannot be negative.
2. Evaluate the function for all possible input values in its domain.
3. The range of the function is the set of all output values that we obtained in step 2.

Classifying a function as an injective function, a surjective function, both or neither:

• An injective function (also known as a one-to-one function) is a function where each input value corresponds to a unique output value.
• A surjective function (also known as an onto function) is a function where every element in the codomain is mapped to by at least one element in the domain.
• A bijection is a function that is both injective and surjective.

Definition of a bijection:

A bijection is a function f between two sets A and B such that every element of A is mapped to exactly one element of B, and every element of B is mapped to by exactly one element of A.

Examples:

• The function f(x) = x^2 is not injective, because the input values -1 and 1 both map to the output value 1.
• The function f(x) = x^2 is not surjective, because the output value 2 is not mapped to by any input value.
• The function f(x) = 2x is injective, because each input value corresponds to a unique output value.
• The function f(x) = 2x is surjective, because every element in the codomain (which is the set of all real numbers) is mapped to by at least one element in the domain (which is also the set of all real numbers).
• The function f(x) = 2x is a bijection, because it is both injective and surjective.

I hope this helps!

A relation R between two sets A and B is a subset of the Cartesian product of A and B.

In other words, a relation R is a set of ordered pairs (a, b) where a is an element of A and b is an element of B.

For example, if A = {1, 2, 3} and B = {4, 5}, then the following set is a relation between A and B:

R = {(1, 4), (2, 5), (3, 4)}

This relation represents the fact that the elements 1 and 4 are related, the elements 2 and 5 are related, and the elements 3 and 4 are related.

Data tables as relations

Data tables can be represented as relations.

In a data table, each row represents a tuple, and each column represents an attribute. The tuples in the data table are related to each other by the attributes in the table.

For example, the following data table represents a relation between the sets of students and courses:

Student	Course
Alice	Math
Bob	Computer Science
Carol	English
David	Math

This data table represents the fact that Alice is enrolled in the Math course, Bob is enrolled in the Computer Science course, Carol is enrolled in the English course, and David is enrolled in the Math course.

I hope this helps!

Why is a number irrational?

Why is 2

2 an irrational number?

2 2 is an irrational number because it cannot be expressed as a fraction of two integers.

To prove this, we can use the following proof by contradiction:

1. Assume that 2 2 can be expressed as a fraction of two integers, p and q, such that p and q are relatively prime (meaning that they have no common factors other than 1).
2. We can then square both sides of the equation, which gives us:

4 = p2/q2

3. Multiplying both sides of the equation by q2, we get:

4q2 = p2

4. This means that p2 is even.
5. Since the square of an odd number is always odd, this means that p must be even.
6. Let p = 2k for some integer k.
7. Substituting this into the equation 4q2 = p2, we get:

4q2 = (2k)2 = 4k2

8. Dividing both sides of the equation by 2k, we get:

2q2 = 2k

9. This means that q is even.
10. But this is a contradiction, because we assumed that p and q were relatively prime, and now we have shown that they are both even.

Therefore, our original assumption must be false, and 2 2 cannot be expressed as a fraction of two integers.

Why can't every number be expressed in p/q form?

Not every number can be expressed in p/q form because the set of rational numbers is countable, while the set of real numbers is uncountable.

This means that there are more real numbers than rational numbers.

Another way to think about it is that the rational numbers are a subset of the real numbers. This means that every rational number is also a real number, but not every real number is a rational number.

Some examples of real numbers that are not rational numbers include:

• π
• e
• √2

These numbers cannot be expressed as fractions of two integers.

I hope this helps!

1. Argue why the set of prime numbers is infinite.

2. Identify if a given number is prime or not.

Why is the set of prime numbers infinite?

The set of prime numbers is infinite because there is no largest prime number.

To prove this, we can use the following proof by contradiction:

1. Assume that there is a largest prime number, p.
2. Consider the following number:

n = p! + 1

3. This number is not divisible by any prime number less than or equal to p, because it leaves a remainder of 1 when divided by any such number.
4. Therefore, n is either prime itself, or it is divisible by a prime number greater than p.
5. In either case, we have found a prime number that is greater than p, which contradicts our assumption that p was the largest prime number.

Therefore, our original assumption must be false, and there cannot be a largest prime number.

How to identify if a given number is prime or not

There are a few different ways to identify if a given number is prime or not. One simple way is to check if the number is divisible by any prime number less than or equal to its square root.

For example, to check if the number 17 is prime, we would check if it is divisible by any prime number less than or equal to the square root of 17, which is 4. Since 17 is not divisible by any of these prime numbers, we can conclude that it is a prime number.

Another way to check if a number is prime is to use the Miller-Rabin primality test. This test is more efficient than the simple method described above, but it is also more complex.

The Miller-Rabin primality test is a probabilistic test, which means that it cannot guarantee to give a correct answer with 100% certainty. However, it is very accurate, and it is the most commonly used primality test in practice.

There are also a number of other primality tests that can be used, but they are all less efficient or less accurate than the Miller-Rabin primality test.

I hope this helps!

Finding the domain and range of a function

The domain of a function is the set of all possible input values for the function. The range of a function is the set of all possible output values for the function.

To find the domain and range of a function, we can use the following steps:

1. Identify any restrictions on the input values of the function. For example, if the function contains a square root radical, then the input value cannot be negative.
2. Evaluate the function for all possible input values in its domain.
3. The range of the function is the set of all output values that we obtained in step 2.

Maximum and minimum values of a function

The maximum value of a function is the largest possible output value of the function. The minimum value of a function is the smallest possible output value of the function.

To find the maximum and minimum values of a function, we can use the following steps:

1. Find the critical points of the function. Critical points are the points where the derivative of the function is equal to zero or undefined.
2. Evaluate the function at the critical points and at the endpoints of its domain.
3. The maximum value of the function is the largest of all the values that we obtained in step 2. The minimum value of the function is the smallest of all the values that we obtained in step 2.

Local maxima and minima

A local maximum of a function is a point in the domain of the function where the function is greater than or equal to all of its neighbors. A local minimum of a function is a point in the domain of the function where the function is less than or equal to all of its neighbors.

To find the local maxima and minima of a function, we can use the following steps:

1. Find the critical points of the function.
2. Evaluate the first derivative of the function at each critical point.
3. If the first derivative is positive at a critical point, then the function is increasing at that point, and the point is a local minimum. If the first derivative is negative at a critical point, then the function is decreasing at that point, and the point is a local maximum.

Identifying whether one function grows faster than another

To identify whether one function grows faster than another, we can use the following steps:

1. Find the limit of the ratio of the two functions as the input value approaches positive or negative infinity.
2. If the limit is equal to infinity, then the first function grows faster than the second function. If the limit is equal to zero, then the second function grows faster than the first function. If the limit is equal to a non-zero finite value, then the two functions grow at the same rate.

For example, consider the following two functions:

f(x) = x^2
g(x) = x^3

As the input value x approaches positive infinity, the limit of the ratio of the two functions is equal to infinity. This means that the function g(x) grows faster than the function f(x).

I hope this helps!