Introduction to Sets

Set is a collection of objects but in mathematics, the definition of the set is one of the most strange definitions. It is a way in mathematics that allows us to put similar objects together. Does mathematics have a tool that can collect families or similar mathematical objects together? A set is such an object. Want to know how? Let us see below!

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Sets

Any well-defined collection of mathematical objects can form a set. These objects could be anything – from people’s names to their ages/likes /dislikes; entities from simple number system to complex scientific data; from outcomes of a single dice roll or a coin toss to such experiments repeated 100s or 1000s of times.

The ONLY condition which is to be kept in mind is that the entities or objects must be related through the same rule. For example:

• Collection of the names of the freedom fighters of India.
• Family of all natural numbers/whole numbers/odd numbers/even numbers/rational numbers/integers/real numbers.
• A group of possible outcomes of a dice roll or a coin toss.
• Collection of crucial data gathered by ISRO from MOM.
• A collection of day/night temperatures.

And there are many more such examples that will form a collection. The important point to notice here is that the rule that defines or law through which we collect or group objects should be universal.

For example, if I say that a group of all the intelligent scientists, it will contain people who I insist are intelligent. If I ask you to form the same group, it will have different names inside it. We say that the rule “intelligent scientists” is not well-defined.

                                                     Learn Types of Sets  here.

 

 

Video on Sets

Conventions for Sets

The following are the conventions that are used here:

• Sets are usually denoted by a capital letter.
• The elements of the group are usually represented by small letters (unless specified separately.)
• If ‘a’ is an element of ‘A’, or if a “belongs to” A, it is written in the conventional notion by the use of the Greek symbol ϵ (Epsilon) between them – a ϵ A.
• If b is not an element of Set A, b “does not belong to” A is written in the conventional notion by the use of the symbol ϵ (Epsilon with a line across it) between them – a ϵ A.
• Objects, elements, entities, members are all synonymous terms.

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Representations of a Set

Representation of Sets and its elements is done in the following two ways.

Roster Form

In this form, all the elements are enclosed within braces {} and they are separated by commas (,). For example, a collection of all the numbers found on a dice N = {1, 2, 3, 4, 5, 6}. Properties of roster form: –

• The order in which the elements are listed in the Roster form for any Set is immaterial. For example, V = {a, e, i, o, u} is same as V = {u, o, e, a, i}
• The dots at the end of the last element of any Set represent its infinite form and indefinite nature. For example, group of odd natural numbers = {1, 3, 5, …}
• In this form of representation, the elements are generally not repeated. For example, the group of letters forming the word POOL = {P, O, L}

More examples for Roster form of representation are:

• A = {3, 6, 9, 12}
• F = {2, 4, 8, 16, 32}
• H = {1, 4, 9, 16, …, 100}
• L = {5, 25, 125, 625}
• Y = {1, 1, 2, 3, 5, 8, …}

Set Builder Form

In this form, all the elements possess a single common property which is NOT featured by any other element outside the Set. For example, a group of vowels in English alphabetical series.

The representation is done as follows. Let V be the collection of all English vowels, then – V = {x: x is a vowel in English alphabetical series.} Properties of Roster form: –

• Colon (:) is a mandatory symbol for this type of representation.
• After the colon sign, we write the common characteristic property possessed by ALL the elements belonging to that Set and enclose it within braces.
• If the Set doesn’t follow a pattern, its Set builder form cannot be written.

More examples for Set builder form of representation for a Set: –

• D = {x: x is an integer and – 3 < x < 19}
• O = {y: y is a natural number greater than 5}
• I = {f: f is a two – digit prime number less than 1000}
• R = {s: s is a natural number such that sum of its digits is 4}
• X = {m: m is a positive integer < 40}

Thus, these were some important points on Sets, what they are, how they are represented mathematically and the related properties.

                                                         Learn Operations on Sets here in detail.

Solved Examples For You

Q1: Write the statements of representation of sets for an unbiased roll for a dice.

Solution:

• In Roster form – A = {1, 2, 3, 4, 5, 6}
• In Set Builder form – A = {x: x is natural number ≤ 6}

Q2: Write the statement of representation of set for a Fibonacci series in Roster form.

Solution: In Roster form, the Fibonacci series can be represented as: A = {1, 1, 2, 3, 5, 8, 13…}. Fibonacci series is a special category series that gets its next number by adding the previous two numbers.

Q. What is a set in Mathematics?

When we look at sets in Mathematics, we see it is a well-defined collection comprising of various objects. For instance, the number 2, 4 and 6 are very different from each other when we look at them separately. However, when you consider them in a collective manner, they make up a single set of size three which we write as {2, 4, 6}

Q. What is a proper set?

A proper subset of a set A is a subset of A which does not equal to A. Meaning to say if B is a proper subset of A then all elements of B are in A but A comprises a minimum of 1 element which is not present in B. For instance, if A = {5, 7, 9} then B = {5, 9}is a proper subset of A.

Q. What is an example of a set?

As a set is a collection of different objects that contain common property, an example of a set will be dog, deer, lion and mouse is all animals. So, when you consider them collectively, they are a set.

Q. What is the symbol of a set?

Sets are commonly denoted with a capital letter like A = {1, 2, 3, 4}. A set which contains no element is an empty or null set and we use { } or ∅ to denote this.