Imagine a very haphazard world where no categories are divided in order to memorize and classify things separately, a world like this will be full of chaos and mess, this is why humans prefer to categorize things and classify them in order to neatly understand and remember them. The same case happens in mathematics, studying math involves dealing with a lot of data, and when the data can be grouped, it is preferred to group them and categorize them, hence, Sets come into play.

### Sets

Sets are defined as the collection of well-defined data. In Math, Set is that tool that helps to classify and collect data belonging to the same category, even though the elements used in sets are all different from each other, but they all are similar as they belong to one group. For instance, a set of different outdoor games, say set A= {Football, basketball, volleyball, cricket, badminton} all the games mentioned are different, but they all are similar in one way as they belong to the same group (outdoor games).

Set is denoted as a capital letter, for example, set A, set B, etc., and the elements belonging to the set are denoted as a small letter, and they are kept in curly brackets {}, for example, set A= {a, b, c, d}, as it is clear that a, b, c, d belong to set A, it can be written a ∈ A, does p belong to set A? No. Therefore, it will be written as, p∉ A.

### Representation of Sets

Sets can be represented in two ways, one is known as the Roster form and the other is famous as the Set-Builder form, these two forms can be used to represent the same data, just the style varies in both cases.

**Roster Form**

In Roster Form, the elements are inside {}⇢ Curly brackets. All the elements are mentioned inside and are separated by commas. Roster form is the easiest way to represent the data in groups. For example, the set for the table of 5 will be, A= {5, 10, 15, 20, 25, 30, 35…..}.

**Properties of Roster form of Sets:**

- The arrangement in the Roster form does not necessarily to be in the same order every time. For example, A= {a, b, c, d, e} is equal to A= {e, d, a, c, b}.
- The elements are not repeated in the set in Roster form, for example, the word “apple” will be written as, A= {a, p, l, e}
- The Finite sets are represented either with all the elements or if the elements are too much, they are represented as dots in the middle. The infinite sets are represented with dots in the end.

**Set-Builder Form**

In Set-builder form, elements are shown or represented in statements expressing relation among elements. The standard form for Set-builder, A= {a: statement}. For example, A = {x: x = a^{3}, a ∈ N, a < 9}

**Properties of Set-builder form:**

- In order to write the set in Set- builder form, the data should follow a certain pattern.
- Colons (:) are necessary in Set-builder form.
- After colon, the statement is to be written.

**Order of the Set**

The order of the Set is determined by the number of elements present in the Set. For example, if there are 10 elements in the set, the order of the set becomes 10. For finite sets, the order of the set is finite and for infinite sets, the order of the set is infinite.

### Sample Problems

**Question 1: Determine which of the following are considered as sets and which are not.**

**All even numbers on the number line.****All the good basketball players from class 9th.****The bad performers from the batch of dancers.****All prime numbers from 1 to 100.****Numbers that are greater than 5 and less than 15.**

**Answer: **

Sets are not those bunches or groups where some quality or characteristic comes in the picture. Therefore,

- “All even numbers on the number line” is a set.
- “All the good basketball players from class 9th” is not a Set as “good” is a quality which is involved.
- “The bad performers from the batch of dancers” cannot be a Set since “bad” is a characteristic.
- “All prime numbers from 1 to 100” is a Set.
- “Numbers that are greater than 5 and less than 15” is a Set.

**Question 2: Represent the following information into the Roster form.**

**All Natural numbers.****Numbers greater than 6 and less than 3.****All even numbers from 10 to 25.**

**Answer:**

The Roster form for the above information,

- Set A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11……}
- Set B = {} ⇢ Null set, since there are no numbers greater than 6 and less than 3.
- Set C = {10, 12, 14, 16, 18, 20, 22, 24}

**Question 3: Express the given information in the Set-Builder form.**

**Numbers that are greater than 10 and less than 20.****All Natural numbers greater than 25.****Vowels in English Alphabet.**

**Answer: **

The Set-Builder form for the above information,

- A = {a: a∈ N and 10 < a < 20}
- B = {b: b∈ N and b > 25}
- C = {c: c is the vowel of English Alphabet}

**Question 4: Convert the following Sets given in Roster form into Set -Builder form.**

**A = {b, c, d, f, g, h, j, k, l, m, n, p, q, r, s, t, v, w, x, y, z}****B = {2, 4, 6, 8, 10}****C = {5, 7, 9, 11,13, 15, 17, 19}**

**Answer: **

The Set- builder form for the above Sets,

- A = {a: a is a consonant of the English Alphabet}
- B = {b: b is an Even number and 2 ≤ b ≤10}
- C = {c: c is an odd number and 5 ≤ c ≤ 19}

**Question 5: Give an example of the following types of Sets in both Roster form and Set-builder form.**

**Singular Set.****Finite Set.****Infinite Set.**

**Solution:**

The Examples can be taken as per choice since there can be a infinite number of examples for any of the above Sets,

- Singular Set
Roster Form: A = {2}

Set- builder form: A= {a: a∈N and 1<a<3}

- Finite Set
Roster Form: B = {0,1, 2, 3, 4, 5}

Set-builder form: B = {b: b is a whole number and b<6}

- Infinite Set
Roster Form: C = {2, 4, 6, 8, 10, 12, 14, 16…..}

Set- builder form: C= {c: c is a Natural and Even number}

**Question 6: What is the order of the given sets,**

**A = {7, 14, 21, 28, 35}****B = {a, b, c, d, e, f, g….x, y, z}****C = {2, 4, 6, 8, 10, 12, 14……}**

**Answer:**

The order of the set tells the number of element present in the Set.

- The order of Set A is 5 as it has 5 elements.
- The order of set B is 26 as the English Alphabet have 26 letters.
- The order of set C is infinite as the set has the infinite number of elements.

**Question 7: Express the given Sets in Roster form,**

**A = {a: a = n/2, n ∈ N, n < 10}****B = {b: b = n**^{2}**, n ∈ N, n ≤ 5}**

**Answer:**

Representing the above Set-builder sets in Roster form,

- A = {1/2, 1, 3/2, 2, 5/2, 3, 7/2, 4, 9/2}
- B = {1, 4, 9, 16, 25}