All students of mathematics should have a good knowledge about set concept. It is essential for learning other topics like geometry, sequence, probability etc.,
Sets and their representations
• Let us first see an example. It can be written in 5 steps:
1. Consider the following group:
The group which contains all odd natural numbers which are less than 10.
• Obviously, that group will contain 1, 3, 5, 7 and 9
2. It is a well-defined group.
• It is well-defined because, doubts of any kind will not arise while we form that group.
3. No member can be excluded.
• None of the numbers 1, 3, 5, 7 and 9 can be excluded from the group
• If any of those numbers is/are excluded, the definition ‘all odd natural numbers which are less than 10’ will become invalid.
4. No other member can be included.
• No number other than 1, 3, 5, 7 and 9 must be included in the group.
• If any other number is included, the definition ‘all odd natural numbers which are less than 10’ will become invalid.
5. So indeed it is a well-defined group.
Let us see another example. It can be written in 5 steps:
1. Consider the following group:
The group which contains all vowels in the English alphabet.
• Obviously, that group will contain a, e, i, o and u
2. It is a well-defined group.
• It is well-defined because, doubts of any kind will not arise while we form that group.
3. No member can be excluded.
• None of the letters a, e, i, o and u can be excluded from the group.
• If any of those letters is/are excluded, the definition ‘all vowels in the English alphabet’ will become invalid.
4. No other member can be included.
• No letter other than a, e, i, o and u must be included in the group.
• If any other letter is included, the definition ‘all vowels in the English alphabet’ will become invalid.
5. So indeed it is a well-defined group.
Let us see one more example. It can be written in 5 steps:
1. Consider the following group:
The group which contains the solution of the equation x2 + x - 6 = 0
• Obviously, that group will contain 2 and -3.
2. It is a well-defined group.
• It is well-defined because, doubts of any kind will not arise while we form that group.
3. No member can be excluded.
• None of values 2 and -3 can be excluded from the group.
• If any of those values is/are excluded, the definition ‘solution of the equation x2 + x - 6 = 0’ will become invalid.
4. No other value can be included.
• No value other than 2 and -3 must be included in the group.
• If any other value is included, the definition ‘solution of the equation x2 + x - 6 = 0’ will become invalid.
5. So indeed it is a well-defined group.
Let us see an example of a group which is not well-defined. It can be written in 3 steps:
1. Consider the following group:
The group which contains the best ten novels in the world.
2. This group is not well-defined because, a novel which one person considers as very good may fail to impress another person.
• So different individuals will be forming different groups.
3. Indeed, such a group is not well-defined.
| ◼ Now we can write the definition of set: A set is a well-defined collection of objects. |
• So the well-defined groups that we saw above can be referred to as:
♦ The set of all odd natural numbers which are less than 10.
♦ The set of all vowels in the English alphabet.
♦ The set of all solution of the equation x2 + x - 6 = 0
• We see that, inside a set, there will be some well-defined objects.
• Those objects can be referred to, in any one of the three different ways given below:
♦ Elements of the set.
♦ Objects of the set.
♦ Members of the set.
◼ Sets are denoted by upper case letters A, B, C, X, Y, Z etc.,
◼ Elements are denoted by lower case letters a, b, c, x, y, z etc.,
• If ‘a’ is an element of set A, we say that: a belongs to A.
♦ The Greek letter epsilon is used to denote ‘belongs to’
✰ So we write: a ∈ A.
✰ And read it as ‘a belongs to A’.
• If ‘b’ does not belong to A,
✰ We write: b ∉ A.
✰ And read it as ‘b does not belong to A’.
◼ An example:
• Let V be the set of vowels in English alphabet.
✰ We can write: i ∈ V.
✰ And read it as ‘i belongs to V’.
• We know that 'd' is not a vowel.
✰ We can write: d ∉ V.
✰ And read it as ‘d does not belong to V’.
◼ Another example:
• The factors of 30 are: 1, 2, 3, 5, 6, 10, 15 and 30.
• Let us make a set P which contains all the prime factors of 30.
♦ Then the elements in that set will be: 2, 3 and 5.
✰ We can write: 5 ∈ P.
✰ And read it as ‘5 belongs to P’.
• We know that '15' is not an element of P.
✰ We can write: 15 ∉ P.
✰ And read it as ‘15 does not belong to P’.
Let us see some predefined sets which we will be using very often in our math classes.
1. We know that the natural numbers are: 1, 2, 3, 4, . . .
♦ The set containing all the natural numbers is denoted as N.
✰ We can write: 5 ∈ N, 7 ∈ N, 235 ∈ N etc.,
✰ Also we can write: 1⁄2 ∉ N, -5 ∉ N, √2 ∉ N etc.,
2. We know that the integers are: . . . -4, -3, -2, -1, 0, 1, 2, 3, . . .
• Integers do not contain decimal or fractional part.
♦ The set containing all the integers is denoted as Z
✰ We can write: -7 ∈ Z, -5 ∈ Z, 235 ∈ Z etc.,
✰ Also we can write: 1⁄2 ∉ Z, -0.35 ∉ Z, √2 ∉ Z etc.,
3. We know that rational numbers are those numbers which can be written in the form p⁄q, where p and q are integers and q is not equal to zero.
♦ The set containing all the rational numbers is denoted as Q
• All integers can be written in the form p⁄q. So they are rational numbers.
✰ We can write: -7 ∈ Q, -5 ∈ Q, 235 ∈ Q etc.,
• All fractions are already in the form p⁄q. So they are rational numbers.
✰ We can write: 1⁄2 ∈ Q, -2⁄3 ∈ Q, 3⁄100 ∈ Q etc.,
• All terminating decimals can be written in the form p⁄q. So they are rational numbers.
♦ Example: 0.35, -0.215 etc., are terminating decimals.
✰ We can write: -7 ∈ Q, -5 ∈ Q, 235 ∈ Q etc.,
• All non-terminating but repeating decimals can be written in the form p⁄q. So they are
rational numbers.
♦ Example: 0.3333..., -0.1111... etc., are non-terminating but repeating decimals.
♦ Recall that: 0.3333... = 1⁄3, -0.1111... = -1⁄9
✰ We can write: 0.3333... ∈ Q, -0.1111... ∈ Q etc.,
4. Non-terminating and non-repeating decimals cannot be written in the form p⁄q. So they are not rational numbers.
♦ Example: 3.14159..., 1.41425... etc., are non terminating and non repeating.
♦ Recall that: 3.14159... = π, 1.41425... = √2
✰ We can write: π ∉ Q, √2 ∉ Q, √5 ∉ Q etc.,
• Such numbers are called irrational numbers.
♦ The set containing all irrational numbers is denoted as T
✰ We can write: π ∈ T, √2 ∈ T, √5 ∈ T etc.,
5. We know that real numbers are those numbers which can be marked on a number line.
• It is easy to mark natural numbers and integers on a number line.
◼ What about rational numbers ?
• Consider 1⁄3 as an example. It can be marked as follows:
♦ On the number line, divide the portion between 0 and 1 into three equal parts.
♦ The end of the first part will be the position of 1⁄3
◼ What about irrational numbers ?
• In our earlier math classes, we have already seen how to mark √2, √5 √7 etc., (See fig.22.4).
◼ So we can write:
All natural numbers, integers, rational numbers and irrational numbers can be marked on a number line.
◼ In other words:
All natural numbers, integers, rational numbers and irrational numbers are real numbers.
♦ The set containing all the real numbers is denoted as R.
✰ We can write: 2 ∈ R, -0.34 ∈ R, 0.3333... ∈ R, √2 ∈ R etc.,
6. We have seen that Z contains all integers only.
• But if we decide to make a set with positive integers only, then that set will be denoted as Z+
7. We have seen that Q contains all rational numbers only.
• But if we decide to make a set with positive rational numbers only, then that set will be denoted as Q+
8. We have seen that R contains all real numbers only.
• But if we decide to make a set with positive real numbers only, then that set will be denoted as R+
Solved example 1.1
Which of the following are sets ? Justify your answer.
(i) The collection of all the months of a year beginning with the letter J.
(ii) The collection of ten most talented writers of India.
(iii) A team of eleven best-cricket batsmen of the world.
(iv) The collection of all boys in your class.
(v) The collection of all natural numbers less than 100.
(vi) A collection of novels written by the writer Munshi Prem Chand.
(vii) The collection of all even integers.
(viii) The collection of questions in this Chapter.
(ix) A collection of most dangerous animals of the world.
Solution:
Part (i):
• The collection is well-defined.
♦ The objects in the collection are: January, June and July.
✰ None of these three months can be excluded from the collection.
✰ No other month can be included in the collection.
◼ So it is a set.
Part (ii):
• The collection is not well-defined.
♦ A writer whom one person considers as very good may fail to impress another person.
♦ So different individuals will be forming different collections.
◼ So it is not a set.
Part (iii):
• The team is not well-defined.
♦ A player whom one person considers as very good may fail to impress another person.
♦ So different individuals will be forming different teams.
◼ So it is not a set.
Part (iv):
• The collection is well-defined.
♦ The objects in the collection are: All the boys in the class.
✰ None of those boys can be excluded from the collection.
✰ No other boy can be included in the collection.
◼ So it is a set.
Part (v):
• The collection is well-defined.
♦ The objects in the collection are: 1, 2, 3, 4, ... , 99
✰ None of those numbers can be excluded from the collection.
✰ No other number can be included in the collection.
◼ So it is a set.
Part (vi):
• The collection is well-defined.
♦ The objects in the collection are: Novels written by the writer Munshi Prem Chand.
✰ None of his novels can be excluded from the collection.
✰ No other novel can be included in the collection.
◼ So it is a set.
Part (vii):
• The collection is well-defined.
♦ The objects in the collection are: . . . -8, -6, -4, -2, 2, 4, 6, . . .
✰ None of those numbers can be excluded from the collection.
✰ No other number can be included in the collection.
◼ So it is a set.
Part (viii):
• The collection is well-defined.
♦ The objects in the collection are: All the questions in this chapter.
✰ None of those questions can be excluded from the collection.
✰ No other question can be included in the collection.
◼ So it is a set.
Part (ix):
• The collection is not well-defined.
♦ An animal which one person considers as 'very dangerous' may be considered as 'not so dangerous' by another person.
♦ So different individuals will be forming different collections.
◼ So it is not a set.
In the next
section, we will see the two methods for representing a set.
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