The Language of Mathematics and Symbols

Summary

This script explores the concept of language, including the learning process of alphabets and grammar. It highlights a 2012 Swedish study showing that learning a new language enhances cortical thickness, beneficial for cognitive functions like multitasking and memory. The script also delves into the idea of mathematics as a language, using symbols to communicate quantitative relationships. It explains set theory basics, including subsets, proper subsets, complements, union, and intersection, using set notation and Venn diagrams to illustrate these concepts.

Takeaways

• 📚 Learning a new language enhances cortical thickness, which is associated with neurons responsible for thought, language consciousness, and memory.
• 🌐 Language acquisition involves learning grammar and structures, which are similar to learning new symbols in a language.
• 🧠 Language learning can improve cognitive functions such as multitasking, problem-solving, and memory, even in tasks unrelated to language.
• 💡 The cognitive benefits of language learning may help counteract the effects of degenerative diseases like dementia and Alzheimer's.
• 🔢 Mathematics is considered a language, allowing for communication through symbols and operations.
• 📘 In mathematics, as in language, we start with basic symbols and progress to more complex expressions and concepts.
• 📈 Sets in mathematics are collections of distinct objects, often denoted by capital letters, and elements within sets are separated by commas.
• 📋 The 'roster method' is used to specify elements in a set, while ellipsis can be used to indicate a continuation of a pattern in sets with many elements.
• 🚫 Common errors in mathematical expressions include incorrect use of minus signs, which should be written as 'x - 1' instead of '1 - x'.
• 📚 Set notation includes the use of set builder notation to define sets with specific properties, such as all real numbers between 0 and 1.
• 🔍 The universal set contains all possible elements, and the complement of a set includes all elements not in the given set.
• ∩ Union and intersection are set operations that combine or find common elements between sets, respectively.

Q & A

What is the significance of learning a new language according to the 2012 Swedish MRI study?

The 2012 Swedish MRI study showed that learning a new language improves cortical thickness, which is a layer of neurons responsible for thought, language consciousness, and memory.

How does learning a new language benefit cognitive abilities?

Learning a new language can improve multitasking, problem-solving, and memory, even when the task at hand has nothing to do with language. It can also help ward off the effects of degenerative diseases like dementia and Alzheimer's.

What are the two main areas of the brain typically associated with language acquisition and storage?

The two main areas are Broca's area, responsible for speech production and articulation, and Wernicke's area in the left temporal lobe, associated with language development and comprehension.

Why is mathematics considered a language?

Mathematics is considered a language because it allows people to communicate with each other using symbols and structures, much like spoken languages do.

What is the common mistake made when translating 'one less than a number' into a mathematical expression?

The common mistake is writing it as '1 - x' instead of the correct 'x - 1'.

What is a set in mathematics?

A set in mathematics is defined as a collection of distinct objects, often named using the capital letter of the English alphabet.

What is the 'roster method' in specifying elements of a set?

The 'roster method' is the specification of elements in a set by enumerating them within braces and separating them with commas.

How is the set of all positive integers represented?

The set of all positive integers can be represented using an ellipsis to indicate the continuation of the pattern, such as {..., -3, -2, -1, 0, 1, 2, ...}.

What is the set builder notation and how is it used?

The set builder notation is used to describe a set by specifying the properties that its elements must satisfy, using the format {x | condition}, where the bar means 'such that'.

What is the difference between a subset and a proper subset?

A subset (A ⊆ B) means all elements of A are in B, but not necessarily all elements of B are in A. A proper subset implies that all elements of A are in B, and A is strictly smaller than B (A ⊂ B).

What is the universal set and how is it related to the complement of a set?

The universal set contains all possible elements under consideration. The complement of a set A (written as A') contains all elements in the universal set that are not in A.

How is the union of two sets represented and what does it include?

The union of sets A and B is represented by A ∪ B and includes all elements that belong to A, or to B, or both.

What is the intersection of two sets and how is it denoted?

The intersection of sets A and B, denoted by A ∩ B, includes only the elements that are common to both A and B.

Can you provide an example of finding the intersection of a set with the complement of the union of two other sets?

Given sets A, B, and C within a universal set, you would first find the union of B and C, then find its complement within the universal set, and finally find the intersection of this complement with set A.

Outlines

00:00

📚 Language Learning and Brain Enhancement

This paragraph discusses the universality of language learning from childhood and the impact of learning new languages on brain structure, specifically mentioning a 2012 Swedish MRI study that found learning a new language improves cortical thickness. It highlights the areas of the brain associated with language acquisition and storage, such as Broca's and Wernicke's areas, and the cognitive benefits of language learning, including improved multitasking, problem-solving, and memory. The paragraph also introduces the concept of math as a language, drawing a parallel between linguistic and mathematical symbols and their progression in complexity.

06:33

🔢 Understanding Mathematical Language and Notation

The second paragraph delves into the precision and power of mathematical language, emphasizing the importance of correct notation and the common pitfalls, such as the proper way to express 'one less than a number.' It introduces the concept of sets in mathematics, explaining how they are defined, named, and represented using the roster method and ellipsis. The paragraph also explores different types of sets, such as the set of all positive integers, negative integers, and integers, and how to express sets containing real numbers between specific bounds using set-builder notation and compound inequalities.

11:34

📐 Set Theory: Subsets, Supersets, and Complements

This paragraph focuses on the fundamental concepts of set theory, including subsets, proper subsets, supersets, and the complement of a set. It clarifies the correct way to denote elements and subsets, using braces to indicate sets and explaining the difference between an element and a subset. The paragraph also discusses the universal set and how to find the complement of a set within it, providing examples to illustrate these concepts.

16:35

🤝 Set Operations: Union, Intersection, and Complements

The final paragraph explores set operations, specifically union and intersection, explaining how to combine elements distinctively in a union and find common elements in an intersection. It uses a Venn diagram to visually represent these operations and provides an example of finding the intersection of set A with the complement of the union of sets B and C within a universal set of positive integers from 1 to 12. The paragraph concludes with the correct identification of the resulting set after performing these operations.

Keywords

💡Language Acquisition

Language acquisition refers to the process of learning a new language. In the video, it is discussed in the context of how learning a new language can improve cognitive abilities such as cortical thickness, which is associated with thought, language consciousness, and memory. The script emphasizes the cognitive benefits of language learning, such as enhanced multitasking, problem-solving, and memory.

💡Alphabet

An alphabet is a set of letters used to write a language. The script mentions that most people learn the symbols of the alphabet to represent the basic sounds of a language. The alphabet is foundational to language learning and is used as a metaphor to introduce the concept of mathematical symbols in the video.

💡Grammar

Grammar refers to the set of structural rules governing the composition of sentences, phrases, and words in a language. The video script discusses grammar in the context of language learning, indicating that as we learn a language, we also learn its grammar, which involves new symbols or combinations of symbols that form words.

💡Broca's Area

Broca's area is a region in the brain associated with speech production and articulation. The script mentions Broca's area as one of the areas involved in language acquisition and storage, highlighting its role in the cognitive process of language.

💡Wernicke's Area

Wernicke's area is a region in the brain's left temporal lobe associated with language development and comprehension. The video script identifies Wernicke's area as another key area for language processing, emphasizing its importance in understanding language.

💡Cognitive Boost

A cognitive boost refers to an improvement in mental functions such as memory, attention, and problem-solving. The script discusses the cognitive boost that can result from learning new languages, which can help ward off the effects of degenerative diseases like dementia and Alzheimer's.

💡Mathematics as a Language

The video script presents the concept of mathematics as a language, emphasizing that it allows people to communicate in a precise and structured way, similar to verbal languages. It draws parallels between the learning of mathematical symbols and the learning of alphabets in verbal languages.

💡Sets

In mathematics, a set is a collection of distinct objects, often named using capital letters of the alphabet. The script explains sets as a fundamental concept in mathematics, using them to illustrate the notation and operations associated with set theory, such as elements, subsets, and the roster method.

💡Roster Method

The roster method is a way of specifying the elements of a set by listing them within curly braces and separating them with commas. The video script uses the roster method as an example of how to denote sets and their elements in mathematical notation.

💡Set Builder Notation

Set builder notation is a way of defining a set by stating the properties that its elements must satisfy, often using a variable and a condition. The script introduces set builder notation as a method to express sets with specific criteria, such as all real numbers between 0 and 1.

💡Empty Set

The empty set, denoted by a pair of curly braces with nothing between them, is a set with no elements. The script mentions the empty set as a concept in set theory, illustrating the notation and the concept of a set that contains no elements.

💡Natural Numbers

Natural numbers are the set of positive integers starting from 1, often denoted by the letter 'N'. The script refers to natural numbers as part of the set theory discussion, indicating that they are a fundamental concept in mathematics.

💡Integers

Integers are a set of numbers that include positive numbers, negative numbers, and zero, often denoted by the letter 'Z'. The script explains integers as a set that forms part of the number system in mathematics, including their notation and the origin of the notation 'Z' from the German word 'Zahlen'.

💡Subset

A subset is a set where all of its elements are contained in another set, known as the superset. The script discusses the concept of subsets in the context of set theory, explaining how one set can be a subset of another and providing examples from the script.

💡Complement of a Set

The complement of a set is the set of all elements in the universal set that are not in the given set, denoted by an apostrophe. The script explains the complement as a set operation, illustrating how to find the complement of a set within a universal set.

💡Union of Sets

The union of sets is a set containing all elements that belong to the first set, the second set, or both, denoted by the symbol 'U'. The script describes the union operation, showing how to combine elements from two sets to form a union.

💡Intersection of Sets

The intersection of sets is a set containing only the elements that are common to two or more sets, denoted by an inverted 'U'. The script explains the intersection operation, providing an example of how to find common elements between two sets.

Highlights

Learning a new language enhances cortical thickness, which is associated with neurons responsible for thought, language consciousness, and memory.

Language learning is not merely muscle work but can be challenging, especially later in life, with significant cognitive benefits.

Broca's area is responsible for speech production and articulation in language acquisition.

Wernicke's area in the left temporal lobe is associated with language development and comprehension.

Learning new languages can improve multitasking, problem-solving, and memory, even in non-language related tasks.

Cognitive boost from language learning can help mitigate the effects of degenerative diseases like dementia and Alzheimer's.

Mathematics is considered a language, allowing for communication through symbols and operations.

Philosophers have noted that the laws of nature are written in the language of mathematics.

In mathematics, symbols represent quantities and operations, building upon foundational symbols as one grows.

Sets in mathematics are collections of distinct objects, often named with capital letters from the English alphabet.

Elements of a set are specified using the roster method, listing them within braces and separated by commas.

Ellipsis in set notation indicates a pattern continuation of elements, simplifying large set representation.

The set builder notation is used to define sets with specific properties, such as all real numbers between 0 and 1.

An empty set is represented by a pair of braces, indicating no elements.

The set of natural numbers (N) and integers (Z) are denoted by specific symbols, representing positive integers and all integers, respectively.

Subsets and proper subsets are defined by the inclusion of elements, with proper subsets having fewer elements than the set they belong to.

The complement of a set contains all elements in the universal set not found in the given set.

Union of sets combines all unique elements from two sets, while intersection finds the common elements.

A complex set operation example is finding the intersection of a set with the complement of the union of two other sets, demonstrating advanced set theory concepts.

Transcripts

since we were a child we started to speak a language

and most of us learn the symbols called the alphabets to represent the basic

sounds of the language

while some of us have other language as our mother tongue

some of us learned other language because of pop sensation

and incidentally we may also have learned few foreign language phrases

because of our favorite shows in TV.

you may also say

You may also say

nevertheless as we learn language we learn structures called grammars and

often times there involves new set of symbols or even combination of symbols

that form words in a 2012 Swedish MRI study showed that

learning a new language improves cortical thickness which is a layered of

mass of neurons responsible for thought language consciousness and memory

learning a new language is little more than the working of a muscle

it can be extremely challenging especially later in life

but the payoff can be big there are a few areas typically associated with

language acquisition and storage we have broca's area which

is responsible for speech production and articulation

and also vernica's area in the left temporal lobe

associated with language development and comprehension

learning new languages can improve multitasking, problem solving, and memory

even when the task at hand has nothing to do with language

that cognitive boost can even help ward up the effects of degenerative diseases

like dementia and Alzheimer's.

Now, do you consider math as a language?

while some find math like a foreign language clearly mathematics is a

language on its own.

Math is a human language just like

english spanish or chinese because it allows people to communicate

with each other this idea of math as a language isn't

exactly new a great philosopher once said the laws

of nature are written in the language of mathematics.

so if in the english alphabet we began

with the symbols

in mathematics we were introduced with

this symbols so the note quantity and operations.

And as we grow we learn how

to read and understand poetry while in mathematics we learned how to

distinguish more symbols.

And further in English we learned formal ways to

communicate to the point that we are writing researches

and in math we learned more sophisticated symbols

And naturally, language grows.

and that includes more ways to express

quantities in mathematics.

Now let's translate the following into mathematical expressions

equations or inequalities

Take note of the common error that when

we say a number less it should be written as minus

x not x minus.

So when we say one less than a number it's written as

x - 1,

Not 1 - x.

Consider that as a language mathematics

is precise concise and powerful.

one of the areas in mathematics that we learned first that use a lot of symbols

is Sets, which is defined as the collection of distinct objects

in which sets are oftentimes named using the capital letter of the english alphabet.

So when we have S as a set,

and 1, 2, 3, 4, 5 as the objects within that set.

Then 1, 2, 3, 4, and 5

are called elements.

And since 1 is one of the five

elements then 1 is an element of S,

which we may also write as 1 ∈ S.

Just remember that sets are always enclosed in braces

and when we enumerate the elements in the given set and separated in comma

we call this specification of elements as 'the roster method'.

on the other hand if there is an element which is not found in

s for example we have 6, then 6 here is not

an element of s and there are sets which have many elements

and enumerating all of them would be burdensome

so what we may do is just put an ellipsis indicating that the elements

after the last specified elements are already understood

based on the pattern indicated by the preceding elements

thus we can say here that S here is a set of

all positive integers on the other hand suppose we have set T having an ellipsis

followed by -3, -2 and -1.

so this shows that this ellipses are the values

-4, -5, -6 and so on or in other words this set T is simply

the set of all negative integers

and sometimes ellipses may be placed at both

ends of the elements in this case since we have 0

1 and 2 as the preceding elements of the ellipses on the right so we know that

this ellipses are the numbers 3, 4, 5 and so

on and on the left side since we have 0

-1, -2 we know that this ellipses

are the elements -3, -4

-5 and so on in other words this set z is simply the set of

all integers now what if you want to know the set

containing all the real numbers between 0 and 1

including 0 and 1. how are you going to write it in set notation

we may express this set in this notation and this is called as the set

builder notation and the bar that's written after the variable x

means 'such that' so S here indicates that S is a set that contains all x such that

x is greater than or equal to zero and at the same time

x is less than or equal to one now instead of writing the word 'AND' we

may replace it by a set operation and that is the

intersection which shows here that x is a number that

is greater than or equal to zero and at the same time it's less than or

equal to one which you may also express

in 'compound inequality' form,

which shows that this x is greater than

or equal to zero and at the same time still it's less

than or equal to one now i'm sure you have already

encountered an empty set written in this simple or simply a pair

of braces you also have encountered the set of

natural numbers denoted by N which is simply all the positive

integers and the set of integers which is denoted

by Z which are composed of the negative

integers the positive integers and 0.

Do you know that the set of integers is often denoted by Z?

Which came from the German word Zahlen, which basically means numbers.

now given the following set will you able to describe this

Now on the next slide, there will be descriptions. Will you able to write it

in set notation?

Before we move on could you identify why these are incorrect?

the reason why this is incorrect is simply because 2 here is written

in braces thus 2 here is a set not an element and since

2 is an element of the given set written in braces

we should write this as 2 is a subset of 1 2 and 3.

we may also write this as 2 without braces

but note that if the given element is not enclosed in braces

that that is considered as an element so we write it as two as element

of one two and three on the next one since

one is not enclosed in braces then one here is an element of

one two and three we may also say that one

written in braces is a subset of one two and three

now i'm sure you have encountered the proper subset

and in this case we simply refer to it as

subset and when we say that A is a subset of B it means that

all the elements in A are in B but at least one element in B is not in A.

So in the example here we see that this set containing the elements 1, 2, 3

are all found in the given set 1, 2, 3, 4, 5

So we can see here that this set is a

subset of the given set so if this is set A

and this is set B we can see here that A is a subset of B again it's because all

elements of A are in B and at least one element of

B is not in A and when we say A is a subset of B conversely

B is a superset of A next we have the complement of a set

wherein if there's a given set A the complement of set A

written in apostrophe is simply the set that contains

all the elements in the universal set which is

not contained in that given set so if we have this universal set

containing the elements 1, 2, 3, 4, 5, 6, 7, 8, 9, 0

if set A is 1, 4, 5, 6 then the complement of A again are the

elements in the universal set which are not in A

and those are the elements 2 3 7 8 9 and 0. so the complement of

A is 2 3 7 8 9 0 next we have is set B with 1, 2, 3

then it shows that 4 ,5 6, 7, 8, 9 and 0 is

the B complement. Now if C contains 0, 1, 2, 3, 4, 5 ,6 ,7 ,8 , 9,

and those are all the elements in the universal set

then the complement of C is null set. now let's proceed to the union of sets

the union of sets of A and B denoted by the symbol U is the set that

contains all the elements that belong to A

or to B or both so if we have A union B that is simply x such that x

is an element of A or element of B again

or both so. in venn diagram as long as that given element is in A

or B or both then that is an element of A

union B. so if you have 1, 3, 4, 5 as set A.

and 3, 4, 7, 8 as set B

then 1, 3, 4, 5, 7, 8 are the elements of the union of sets A

and B it's simply because 1 is an element of A

5 is an element of A, 7 and 8 are elements of B, 3 and 4 are

elements of both. for the intersection of sets A and

B denoted by the inverted U those are

simply the elements that are common to both A and B so A intersection

B if x is such that x is an element of A

and at the same time is an element of B. so if we have set A that contains 1,3,4,5

and set B that contains 3,4,7,8

then the intersection of A and B is

simply the set that contains 3 and 4. since 3 and 4 are the

common elements just remember that in the union of sets

we simply combine all the elements distinctively

so if there are repeated elements we don't write it again

and in the intersection of sets we simply find the common elements

now given the universal set containing the positive integers from 1 to 12

and given these sets A, B, and C can you find the intersection of

A with the complement of the union of B and C?

to answer this we simply find the union first of B and C.

meaning we're going to combine all the elements of B and C.

and combining these elements we're simply going to have

1 2 3 4 5 6 7 and 8 and when we try to find the

complement of that meaning we're going to find all the

elements in the universal set which are not 1, 2, 3, 4, 5, 6, 7, 8

then those elements are simply 9 10 11 and 12

and the intersection of that set with A

is simply the set that contains the common elements which are 9 and 10