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Set language and notation

Updated: Jun 22, 2021


Dear Secondary Math students, we will be going through Set Language and Notations. This chapter consists of many special and unique symbols which you might not come across. So stay tuned and pay close attention to them!



In this note, you will learn:


1. Use of set language and notations (Union, Intersection, etc.)

Use of set language and notations (Union, Intersection, etc.)


A “set” in mathematics context refers to the collection of object, things or symbols that are clearly defined. Each of the individual object in a set are called elements or members of the set.



Elements can be presented in two ways:

  1. By listing the elements

  2. By description

When we list elements of a set, we typical use curly brackets to represent a set. In a description, it is used to represent the phrase, “the set of” and we write the elements/members of a set within the curly brackets.


For example:


Listing Description

{1, 3, 5, 7, 9} The set of odd numbers between 0 to 10

{5, 10, 15, 20, 25, 30} The set of multiples of 5 from 5 to 30

Union of two sets, “A ∪ B”:


The union of two sets is denoted by the symbol, ‘’.


Given that we have two sets, A and B. The union of A and B basically means that the new set must include all the elements that are unique to A, all the elements that are unique to B, and the elements that exist in both A and B.


For example:


Set A consist of the numbers from 1 to 10:

A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}


and Set B consist of numbers from 5 to 15.

B = {5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}


Hence, if we identify the elements that are unique to both sets A and B individually, and the elements that exist in both sets:


A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

B = {5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}


Therefore


A ∪ B = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}

*Note: For union, elements that exist in both sets do not have to be repeated.

Intersection of two sets, “A B”:


The intersection of two sets is denoted by the symbol, ‘∩’.


Given that we have two sets, A and B. The intersection of C and D basically means that the new set must include only the elements that exists in BOTH sets A and B.


For example:

Set A consist of the test results of 5 students from class 1:

A = {58, 66, 72, 40, 87}


Set B consist of the test results of 5 students from class 2:

B = {58, 54, 62, 87, 72}


Hence, if we identify the elements that exist in both sets A and B:


A = {58, 66, 72, 40, 87}

B = {58, 54, 62, 87, 72}


Therefore,


A ∩ B = {58, 72, 82}

Complement of a set, A’:


The complement of a set is denoted by the original set followed by an apostrophe, i.e. The complement of set A is A’.


The complement of a set is basically a set that has all the elements in the universal set, except those that are in the original set A.


For example:

If Ƹ = {a, b, c, d, e, f} and A = {b, d, f},

Therefore, A’ = {a, c, e}

“… is a (proper) subset of …”, or “A ⊂ B”:


Given that we have two sets, A and B. When we say set A is a subset of set B, it is denoted by ‘’, and this basically means that every element in set A also exists in set B.

When we say set A is a proper subset of set B, this basically means that not only every element in set A also exists in set B, set B has MORE elements than set A.


E.g. Given that A = {vowels of the English Alphabet} and B = {letters of the English Alphabet}


This means that set A is a proper subset of set B, or A ⊂ B, since set A contains only “a, e, i, o, u”, but set B contains all the 26 letters of the English alphabet, which has consist of all the elements in set A and has more elements than set A as well.

“… is not a (proper) subset of …”, or “A ⊄ B”:


When a set A is a not subset of set B, it is denoted by ‘’. This basically means that there is at least one element in set A that does not exist in set B.


E.g. Given that A = {vowels of the English Alphabet, 1, 2, 3} and B = {letters of the English Alphabet}


This means that set A is not a subset of set B, or A ⊄ B, because although set A contains “a,e,i,o,u”, which are all elements that exist in set B, which contains all the 26 letters of the English alphabet, but set B does not contain the numbers, “1, 2, 3” unlike in set A.

*Some other important notations in set language that you must know:


Ƹ – represents a universal set. A universal set is a set that contains all the objects or elements and of which all other sets are subsets.


E.g. Given that Ƹ = {1, 2, 3… 998, 999, 1000}, then other sets must only contain elements that are within the universal set i.e. from 1 to 1000


∈ - represents ‘… is an element of …’

E.g. x A, which also means “x is an element of A


∉ - represents ‘…is not an element of …’

E.g. y ∉ A, which also means “y is not an element of A

or { }- represents an empty set. An empty set is basically a set that does not contain any elements at all.


E.g. A = {1, 3, 5, 7}, B = {2, 4, 6, 8}

A ∩ B = { } = ∅


And that’s all for today, students! Math Lobby hopes that after this article, you have a clear understanding on the set language and notations!



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