Representation of a Set

Sets are defined as collections of well-defined data. In Math, a Set is a 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, they are all 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).

Sets can be represented in two ways: Set-Builder form and Roster form.

Let's discuss the above forms in detail with solved examples and Properties.

Roster Form

In Roster Form, the elements are inside { }⇢ Curly brackets. All the elements are mentioned inside and are separated by commas. A 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 have 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 atthe the end.

Set-Builder Form

In Set-builder form, elements are shown or represented in statements expressing relations among elements. The standard form for Set-builder, A= {a: statement}.

For example, A = {x: x = a³, 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 the 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 on Representation of Sets

Question 1: Determine which of the following are considered assets and which are not.

1. All even numbers are on the number line.
2. All the good basketball players are from class 9th.
3. The bad performers are from the batch of dancers.
4. All prime numbers from 1 to 100.
5. 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,

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

Question 2: Represent the following information in the Set-Builder Roster form.

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

Answer:

The Roster form for the above information,

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

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

1. Numbers that are greater than 10 and less than 20.
2. All Natural numbers greater than 25.
3. Vowels in the English Alphabet.

Answer: 

The Set-Builder form for the above information,

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

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

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

Answer: 

The Set- builder form for the above Sets,

1. A = {a: a is a consonant of the English Alphabet}
2. B = {b: b is an Even number and 2 ≤ b ≤10}
3. 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.

1. Singular Set.
2. Finite Set.
3. 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?

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

Answer:

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

1. The order of Set A is 5 as it has 5 elements.
2. The order of set B is 26 as the English Alphabet have 26 letters.
3. 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.

1. A = {a: a = n/2, n ∈ N, n < 10}
2. B = {b: b = n², n ∈ N, n ≤ 5}

Answer:

Representing the above Set-builder sets in Roster form,

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

Practice Problems on Representation of Sets

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

1. All prime numbers less than 50.
2. All the tall people in the class.
3. All even numbers between 30 and 50.
4. Numbers greater than 100 and less than 200.
5. All perfect squares from 1 to 100.

Question 2: Represent the following information in the Set-Builder and Roster form.

1. All whole numbers.
2. Numbers greater than 10 but less than 20.
3. All odd numbers between 1 and 15.
4. All integers from -5 to 5.

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

1. All prime numbers less than 20.
2. All numbers that are divisible by 5 and greater than 20.
3. All natural numbers greater than 50 and less than 100.
4. Vowels in the alphabet (both upper and lower case).

Question 4: Convert the following sets from Roster form into Set-Builder form.

1. A = {1, 3, 5, 7, 9, 11, 13, 15, 17, 19}
2. B = {a, e, i, o, u}
3. C = {2, 4, 6, 8, 10, 12, 14, 16, 18}
4. D = {0, 1, 4, 9, 16, 25}

Question 5: Provide examples of the following types of sets in both Roster form and Set-builder form.

1. Singleton Set.
2. Finite Set.
3. Infinite Set.
4. Universal Set.

Question 6: What is the order of the following sets?

1. A = {2, 4, 8, 16, 32}
2. B = {p, q, r, s, t}
3. C = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
4. D = {1, 2, 3, 4, 5, ..., ∞}

Question 7: Express the given sets in Roster form.

1. A = {a: a is an even number and a < 10}
2. B = {b: b is a perfect square and 0 ≤ b ≤ 25}

Question 8: Determine if the following statements represent valid sets:

1. The set of all red-colored balls in a bag.
2. The set of all green colors in a rainbow.
3. The set of all tall buildings in New York City.
4. The set of all positive integers less than 50.

Conclusion

• The representation of a set is a fundamental concept in mathematics that allows us to describe and manipulate collections of distinct objects or elements.
• The Sets can be represented in various forms, including the roster form, set-builder form and Venn diagrams.
• Understanding these different representations helps in visualizing and solving problems related to the unions, intersections, subsets and other set operations.
• Mastery of set representation is essential for the students and professionals working in the fields like mathematics, computer science and logic where sets form the basis for the more complex concepts.