How to Insert the appropriate Symbol either an Element or not an element to a Set?

Again, the collection of five most renowned mathematicians of the world is not well—defined, because the criterion for determining a mathematician as most renowned may vary from person to person. Thus, it is not a well—defined collection.

We shall say that a set is a well-defined collection of objects.

The following points may be noted:

(i) Objects, elements and members of a set are synonymous terms.

(ii) Sets are usually denoted by capital letters A, B, C, X, Y, Z, etc.

(iii) The elements of a set are represented by small letters a, b, c, x, y, z, etc.

If a is an element of a set A, we say that “a belongs to A” the Greek symbol ∈ (epsilon) is used to denote the phrase ‘belongs to’. Thus, we write aA. If ‘b’ is not an element of a set A, we write bA and read “b does not belong to A”.
Thus, in the set V of vowels in the English alphabet, a∈V but b∉V. In the set P of prime factors of 30, 3∈P but 15∉P. There are two methods of representing a set :

(i) Roster or tabular form

(ii) Set—builder form.

🌈 Sets in Roster or Tabular Form or Listing Method AND Set-Builder Form or Rule Method
The symbols ∈ and ∉
The phrases ’is an element of‘ and ‘is not an element of’ occur so often in discussing sets that the special symbols ∈ and ∉ are used for them. For example, if A={3,4,5,6}, then
3∈A (Read this as “3 is an element of the set A“.)
8∉A (Read this as “8 is not an element of the set A“.)

Describing and naming sets
● A set is a collection of objects, called the elements of the set.
● A set must be well defined, meaning that its elements can be described and listed without ambiguity. For example:

{1,3,5} and {letters of the English alphabet}.

● Two sets are called equal if they have exactly the same elements.

• The order is irrelevant.

• Any repetition of an element is ignored.

● If a is an element of a set S, we write a∈S.
● If b is not an element of a set S, we write b∉S

We read the notation “5∈D” as “5 is an element of D”.
We read the notation “2∉D” as “2 is not an element of D

If B is a set, and x is an object contained in B, we Write xB. If x is not contained in B then we write xB.

Examples.
● 5∈{2,3,5}
● 1∉{2,3,5}
📌 Example 1. State which of the following statements are true and which are false. Justify your answer.
(i) 37∉ {x|x has exactly two positive factors}
(ii) 28∈ {y| the sum of the all positive factors of y is 2y}
(iii) 7, 747∈ {t|t is a multiple of 37}
✍ Solution:
(i) False. Since, 37 has exactly two positive factors, 1 and 37, 37 belongs to the set.
(ii) True. Since, the sum of positive factors of 28

=1+2+4+7+14+28=56=2(28)

(iii) False. 7, 747 is not a multiple of 37.

📌 Ex2. State which of the following statements are true and which are false? Justify your answer.

(i) 35 ∈{x:x has exactly four positive factors}
(ii) 128 ∈{y:the sum of all the positive factors of y is 2y}
(iii) 3 ∉ {x:x4-5x3+2x2-112x+6=0}
(iv) 496 ∉{y:the sum of all the positive factors of y is 2y}

🔑 Answers: (i) true (ii) false (iii) true (iv) false
🌟 Inserting the appropriate symbol either an element or not an element to a set