The Language of Mathematics and Symbols

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