Listing the elements of a set inside a pair of braces, {}. is called roster form. The braces are an essential part of the notation because they identify the contents as at set. For example, {1,2,3} is notation for the set whose elements are 1, 2, and 3. but (1,2,3) and [1,2,3] are not sets because parentheses and brackets do not indicate a set. For a set written in roster form, commas separate the elements of the set. The order in which the elements are listed is not important.

Sets are generally named with capital letters. For example, the name commonly selected for the set of natural numbers or counting numbers is ℕ.

Definition: Natural Numbers

ℕ={1,2,3,4,5,…}
The three dots after the 5, called an ellipsis, indicate that the elements in the set continue in the same manner. An ellipsis followed by a last element indicates that the elements continue in the same manner up to and including the last element. This notation is illustrated in Example 1.

📌 Example 1: Roster Form of a Set
Set A is the set of natural numbers less than or equal to 80. Express the set A in roster form.
✍ Solution:
A={1,2,3,4,…,80}. The 80 after the ellipsis indicates that the elements continue in the same manner up to and including the number 80.

Specifying Sets
There are at least two different ways of specifying sets:
(i) One method of specifying a set is to list all the members of the set between a pair of braces. Thus {1,2,3} represents a set. This method is called “The listing method”.
📌 Example 2:

(a) {3,6,9,12,15} (b) {a,b,c,d}

This method of listing the elements of the set is also known as ‘Tabulation’. In this method the order in which the elements are listed is immaterial, and is used for small sets.

(ii) Another method of defining particular sets is by a description of some attribute or characteristic of the elements of the set. This method is more general and involves a description of the set property.

B={x|x has the property P}

Designates “the set B of all objects ‘x’ such that x has the property P”. This notation is called SetBuilder notation. The vertical bar | is read as “such that”.

📌 Example 3:
(a) C={x|x is a positive Integer greater then 100}.
This is read as “the set of all x is a positive Integer less than 25”.
(b) D={x|x is a complex number}.
[Note]: Repetition of objects is not allowed in a set, and a set is collection of objects without ordering.

📌 Ex4. Specify the set E by listing its elements, where E={whole numbers less than 100 divisible by 16}.
🔑 E={0,16,32,48,64,80,96}.

📌 Ex5. List out the elements of the set “The letters of the word Mississipi”
🔑 {m,i,s,p}

📌 Ex6. Use listing method to express the set

F={x:x=n³, n∈ℕ and x<80}.
🔑 F={1,8,27,64}

📌 Ex7. List the elements of the set

G={x:x is an Integer, -½<x<-9/2}
🔑 G={0,1,2,3,4}

📌 Ex8. Let U={x:x∈ℕ, x<50}, H={x:x²∈ξ}, J={x:x=n², n∈ℕ} and K={x:x is a factor of 36}. List the elements of each of the sets H, J and K.
✍ Solution:
ξ={x:x∈ℕ, x<50}={1,2,3,4,5,…,49} H={x:x²∈U}={1,2,3,4,5,6,7}
J={x:x=n², n∈ℕ, x∈U}={1,4,9,16,25,36,49}
K={x:x is a factor of 36}={1,2,3,4,6,9,12,18,36}

📌 Ex9. Writing each of the following sets by listing their elements between braces.

{xℤ:-2≤x<7}={-2,-1,0,1,2,3,4,5,6} {x∈ℝ:x2=3}={-√3,√3} {x∈ℝ:x2+5x=-6}={-2,-3} {x∈ℤ:|x|<5}={-4,-3,-2,-1,0,1,2,3,4} {x∈ℤ:|6x|<5}={0}

Note that ℤ=the set of all integers. {…,-3,-2,-1,0,1,2,3,…}, and ℝ=the set of real numbers including rational and irrational numbers.

📌 Ex10. List all the elements of the following sets:
(i) L={x:x is an odd natural number}
(ii) M={x:x is an integer; -½<x<9/2} (iii) N={x:x is an integer; x²≤4}
(iv) O={x:x is a letter in the word “LOYAL”}
(v) P={x:x is a month of a year not having 31 days}
(vi) Q={x:x is a consonant in the English alphabet which proceeds k}.
✍ Solution:
(i) L={x:x is an odd natural number}={1,3,5,7,9,…}
(ii) M={x:x is an integer; -½<x<9/2} It can be seen that -½=-0.5 and 9/2=4.5 ∴={0,1,2,3,4} (iii) N={x:x is an integer; x²≤4}
It can be seen that

(-1)²=1≤4; (-2)²=4≤4; (-3)²=9>4
0²=0≤4
1²=1≤4
2²=4≤4
3²=9>4
(iv) O=(x:x is a letter in the word “LOYAL”)={L,O,Y,A}
(v) P={x:x is a month of a year not having 31 days}

={February, April, June, September, November}
(vi) Q={x:x is a consonant in the English alphabet which precedes k}

={b,c,d,f,g,h,j}

🌈Particular Symbols ℕ 𝕎 ℤ ℚ 𝕋 ℝ ℂ Represent Number Sets in Mathematics🌟