A set is denoted by capital letters like “A”, “B”, “C” etc. In choosing a symbol for a set, it is generally convenient to use a capital letter that identifies with the set. For example, it is appropriate to use symbol “V” to represent collection of vowels in English alphabet.
On the other hand, the members or elements of a set are denoted by small letters like “a”,”b”,”c” etc.
Membership of a set is denoted by the symbol “
The set is represented in two ways :
- Roaster form
- Set builder form
Roaster form
All elements of the set are listed with a comma (“,”) in between and the listing itself is enclosed within braces “{“ and “}”. The order or sequence of elements within the set is not important – though desirable.
The set of numbers, which divide 12, is written as :
If a pattern or sequence is easily made out, then we can use ellipsis ("...") to represent continuity of such pattern. This type of representation is particularly useful to represent an infinite set. Clearly, sequence of members in this type of representation is important.
The set of even numbers is written as,
The roaster form is limited in certain circumstance. For example, we can not represent set of real numbers in roaster form. Real numbers is an infinite set, but the elements of this set do not follow a pattern or have a particular sequence. As such, we can not define same with the help of ellipsis.
Every member of the set is unique and distinct. However, we encounter situations in which collection can have repeated elements. For example, the set representing scores of three students can be a set of three numbers one of which is repeated :
We need to reduce such collection as :
Set builder form
Collections are often characterized by a common property. We can, therefore, define members of a set in terms of the common property. However, we need to ensure that objects outside the collection do not have the same common property.
The construction of qualification for the common property is quite flexible. Only thing is that we need to be explicit in what we mean. Generally, we denote the member by a symbol like “x” and then define the membership. Consider the examples :
The roaster equivalents of two sets are :
Can we write set “B” as the one which comprises single digit natural number? Yes. Thus, we can see that there are indeed different ways to define and identify members and hence the flexibility in defining collection.
We should be careful in using words like “and” and “or” in writing qualification for the set. Consider the example here :
Both conditional qualifications are used to determine the collection. The elements of the set as defined above are integers. Thus, the only member of the set is “3”.
Now, let us consider an example, which involves “or” in the qualification,
The member of this set can be elements belonging to either of two sets "A" and "B". The set consists of elements (i) belonging exclusively to set "A", (ii) elements belonging exclusively to set "B" and (iii) elements common to sets "A" and "B".
Example
Problem 1 : A set in roaster form is given as :
Write the set in “set builder form”.
Solution : We see here that we are dealing with natural numbers. The numerators are square of natural numbers in sequence. The number in denominator is one more than numerator for each member. We can denote natural number by “n”. Clearly, if numerator is “
However, this set is not an infinite set. It has exactly three members. Therefore, we need to specify “n” so that only members of the set are exclusively denoted by the above expression. We see here that “n” is greater than 4, but “n” is less than 8. For representing three elements of the set,
We can write the set, now, in the builder form as :
In set builder form, the sequence within the range is implied. It means that we start with the first valid natural number and proceed sequentially till the last valid natural number.
