In the previous section, we saw an introduction to sets. In this section, we will see the two methods for representing a set.
◼ There are two methods for representing a set.
♦ Roster form
✰ This is also known as Tabular form.
♦ Set-builder form
We will first see roster form. Some basic features can be written in 5 steps:
1. In roster form, all elements of the set are clearly written.
2. The elements are separated by commas.
3. The elements are enclosed within braces.
4. The order in which the elements are written is not important.
5. Repeating elements are written only once.
Let us see some examples:
(a) Consider the following set:
• The set of all even positive integers which are less than 7.
♦ Clearly, the elements are 2, 4, and 6.
♦ We must clearly write all those three elements.
♦ We must separate them with commas.
♦ We must enclose them within braces.
• So the set will appear as: {2, 4, 6}
• If we decide to name the set as A, we can write: A = {2, 4, 6}
• Since the order is not important, we can write A = {4, 6, 2} also. Different arrangements like this are allowed.
(b) Consider the following set:
• The set of all natural numbers which completely divide 42
♦ Clearly, the elements are 1, 2, 3, 6, 7, 14, 21, and 42
♦ We must clearly write all those 8 elements.
♦ We must separate them with commas.
♦ We must enclose them within braces.
• So the set will appear as: {1, 2, 3, 6, 7, 14, 21, 42}
• If we decide to name the set as B, we can write: B = {1, 2, 3, 6, 7, 14, 21, 42}
• Since the order is not important, we can write B = {21, 42, 7, 14, 3, 6, 1, 2} also. Different arrangements like this are allowed.
(c) Consider the following set:
• The set of all vowels in English alphabet.
♦ Clearly, the elements are a, e, i, o, and u.
♦ We must clearly write all those 5 elements.
♦ We must separate them with commas.
♦ We must enclose them within braces.
• So the set will appear as: {a, e, i, o, u}
• If we decide to name the set as V, we can write: V = {a, e, i, o, u}
• Since the order is not important, we can write V = {i, o, a, e, u} also. Different arrangements like this are allowed.
(d) Consider the following set:
• The set of all natural odd numbers.
♦ Clearly, the elements are 1, 3, 5, 7, ...
✰ These numbers continue indefinitely. We cannot write them all.
♦ In such cases, we write the first 4 elements, followed by a comma and 3 dots.
♦ We must enclose them within braces.
• So the set will appear as: {1, 3, 5, 7, ...}
• If we decide to name the set as D, we can write: D = {1, 3, 5, 7, ...}
• Here the order is important.
(e) Consider the following set:
• The set of all letters in the word 'SCHOOL'
♦ Clearly, the letters are S, C, H, O, O and L.
✰ We must write 'O' only once
• So the set will appear as: {S, C, H, O, L}
• If we decide to name the set as E, we can write: E = {S, C, H, O, L}
• Since the order is not important, we can write E = {S, H, C, L, O} also. Different arrangements like this are allowed.
Next we will see the set-builder form. It can be explained in 7 steps:
1. We know that, all the elements of a set will possess a common property.
♦ All the elements which possess that property should be included into that set.
✰ None of such elements should be excluded.
♦ Also, objects which do not possess that property should not be included into that set.
2. This gives us an idea to specify a set:
• A set can be specified by writing down the property.
♦ A person who reads the property can easily form the set.
3. Let us see an example:
• The property possessed by every element of a set is this:
It is a vowel of English alphabet.
• A person who reads this property will understand that, elements in the set are: a, e, i, o and u.
4. Another example:
• The property possessed by every element of a set is this:
It is positive even integer less than 10.
• A person who reads this property will understand that, elements in the set are: 2, 4, 6, and 8.
5. Mathematicians have specified the form in which the property is to be written.
• By making such a specification, we can ensure that, people all over the world will follow the same form.
◼ It is called the set-builder form.
6. We can familiarize ourselves with the form through some examples:
Example 1:
• The set V of vowels in English alphabet is written in set builder form as:
V = {x : x is a vowel in English alphabet}
• It is read as:
The set of all x such that x is a vowel of the English alphabet.
♦ The braces stand for: The set of all.
♦ The colon ':' stands for: such that.
• So we can read in the order as shown in fig.1.1 below:
 |
| Fig.1.1 |
Example 2:
 |
| Fig.1.2 |
7. So now we know how to write the set-builder form.
• Note that, ‘x’ is a variable. We know that, a variable can have different values.
• However, the values taken up by ‘x’ should satisfy the property given after the colon ‘:’.
• Remember that, instead of 'x', we can use, 'y', 'z' etc., also.
Let us see the previous examples that we wrote for roster form. We should be able to write each of them in set-builder form also:
(a) The set of all even positive integers which are less than 7.
• In roster form, we wrote: A = {2, 4, 6}
• In set-builder form, this will be:
A = {x : x is an even positive integer less than 7}
(b) The set of all natural numbers which completely divide 42
• In roster form, we wrote: B = {1, 2, 3, 6, 7, 14, 21, 42}
• In set-builder form, this will be:
A = {x : x is a natural number which completely divide 42}
(c) The set of all vowels in English alphabet.
• In roster form, we wrote: V = {a, e, i, o, u}
• In set-builder form, this will be:
V = {x : x is a vowel in English alphabet}
(d) The set of all natural odd numbers.
• In roster form, we wrote: D = {1, 3, 5, 7, ...}
• In set-builder form, this will be:
V = {x : x is a natural odd number}
(e) The set of all letters in the word 'SCHOOL'
• In roster form, we wrote: E = {S, C, H, O, L}
• In set-builder form, this will be:
V = {x : x is a letter in the word 'SCHOOL'}
Solved example 1.2
Write the solution set of the equation x2 + x - 2 = 0 in roster form.
Solution:
• The given equation can be written as: $\mathbf\small{\rm{x^2+x=-2}}$
⇒ $\mathbf\small{\rm{x^2+x+\left(\frac{1}{2} \right)^2=-2+\left(\frac{1}{2} \right)^2}}$
• We take the half of the coefficient of x. We then add it's square to both sides. Thus we get:
$\mathbf\small{\rm{\left(x+\frac{1}{2} \right)^2=\frac{9}{4}}}$
⇒ $\mathbf\small{\rm{\left(x+\frac{1}{2} \right)=\pm\frac{3}{2}}}$
⇒ x = 1, -2
• So the solution set of the given equation in the roster form is: {1, -2}.
Solved example 1.3
Write the set {x : x is a positive integer and x2 < 40}in the roster form.
Solution:
1. The positive integers are: 1, 2, 3, 4,...
• All integers from 1 to 6 will satisfy x2 < 40
2. So the values which x can take are: 1, 2, 3, 4, 5 and 6
• Thus the set in roster form is: {1, 2, 3, 4, 5, 6}
Solved example 1.4
Write the set A = {1, 4, 9, 16, 25, . . .} in set builder form.
Solution:
1. It is clear that, the given set contains infinite number of elements. They are the squares of natural numbers 1, 2, 3, 4, . . .
2. So in set-builder form, it will be:
A = {x : x is square of a natural number}
3. Alternatively, we can write: A = {x : x = n2, where n ∈ N}
Solved example 1.5
Write the set $\mathbf\small{\rm{\{\frac{1}{2},\frac{2}{3},\frac{3}{4},\frac{4}{5},\frac{5}{6},\frac{6}{7}\}}}$ in the set-builder form.
Solution:
1. We see that,
♦ each x is a fraction
♦ All numerators and denominators are natural numbers
• So they belong to the set N
2. If we denote the numerator as n, the denominator will be (n+1)
• So each x is $\mathbf\small{\rm{\frac{n}{n+1}}}$
3. Thus the set-builder form is: {x : $\mathbf\small{\rm{n=\frac{n}{n+1}}}$ and 1 ≤ n ≤ 6}
The link below gives a 4 more solved examples:
In the next
section, we will see the Empty set
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