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 *a*∈*A*. If ‘*b*’ is not an element of a set *A*, we write *b*∉*A* 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:

● 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 *x*∈*B*. If *x* is not contained in *B* then we write *x*∉*B*.

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 2*y*}

(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

(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:x^{4}-5x^{3}+2x^{2}-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