In the previous section,
we saw the basics about subsets. In this section, we will see the subsets of R. Later in this section, we will also see intervals, power sets and the universal set.
• In the first section of this chapter, we saw the details about sets like N, Z, Q, T, R etc., (Details here).
• Based on that, we can write about the subsets of R. It can be written in 4 steps:
1. We can plot all elements of Z on the number line.
• When we make such a plot, all the elements of N will also be present among those Z values.
◼ Consider the reverse:
• We can plot all elements of N on the number line
• When we make such a plot, all the elements of Z will not be present among those N values.
◼ So we can write:
♦ All elements of N are present in Z.
♦ All elements of Z are not present in N.
• Thus we get: N ⊂ Z and Z ⊄ N
2. We can plot all elements of Q on the number line.
• When we make such a plot, all the elements of Z will also be present among those Q values.
◼ Consider the reverse:
• We can plot all elements of Z on the number line.
• When we make such a plot, all the elements of Q will not be present among those Z values.
◼ So we can write:
♦ All elements of Z are present in Q.
♦ All elements of Q are not present in Z.
• Thus we get: Z ⊂ Q and Q ⊄ Z
• But in (1), we saw that, N ⊂ Z
♦ So we can write: N ⊂ Z ⊂ Q
• That means, when we plot Q, all elements of N and Z will also be present among those Q values.
3. We can plot all elements of R on the number line.
• When we make such a plot, all the elements of Q will also be present among those R values.
◼ Consider the reverse:
• We can plot all elements of Q on the number line.
• When we make such a plot, all the elements of R will not be present among those Q values.
◼ So we can write:
♦ All elements of Q are present in R.
♦ All elements of R are not present in Q.
• Thus we get: Q ⊂ R and R ⊄ Q
• But in (2), we saw that, Z ⊂ Q
♦ So we can write: Z ⊂ Q ⊂ R
• Also in (1), we saw that, N ⊂ Z
♦ So we can write: N ⊂ Z ⊂ Q ⊂ R
• That means, when we plot R, all elements of N, Z and Q will also be present among those R values.
4. We know the reason why Q does not contain all the elements of R
• Let us write it:
Set R contains both rational numbers and irrational numbers (T). But Q does not contain irrational numbers
Intervals as subsets of R
This can be explained in 9 steps:
1. Let a and b be two real numbers, with b being greater than a.
♦ a will have a unique position on the number line.
♦ b will also have a unique position on the number line.
• Those positions are shown in fig.1.3(i) below:
 |
| Fig.1.3 |
2. There will be infinite number of points between a and b.
• The portion between a and b is a continuous portion. There are no breaks between a and b.
• So we can say:
The portion between a and b is an interval between a and b in the number line.
3. We can define a set which contains all the points in that interval.
• The points in that set must satisfy two conditions:
(i) The points must lie to the right of a.
(ii) The points must lie to the left of b.
◼ This set can be written in set-builder form as: {y : a < y < b}
That means, y is greater than a and at the same time, less than b.
4. Whether to include a and b in the set:
• y must be greater than a.
♦ So a is not included.
• y must be less than b.
♦ So b is not included.
◼ That means, a and b, which defines the boundaries of the interval, are not included in the set.
◼ So it is called an open interval.
• It is denoted as (a, b).
5. If we include the boundaries also, then it is called a closed interval.
♦ It is denoted as [a, b].
• In set-builder form, it will be: {y : a ≤ y ≤ b}
♦ It is shown in fig.1.3(ii) above.
♦ Both a and b will be present in the set.
6. We can also have intervals which are closed at one end and open at the other.
• (a, b] is open at a and closed at b.
♦ In set-builder form, it will be: {y : a < y ≤ b}
♦ It is shown in fig.1.3(iii) above.
♦ a will not be present in the set but b will be present.
• [a, b) is closed at a and open at b.
♦ In set-builder form, it will be: {y : a ≤ y < b}
♦ It is shown in fig.1.3(iv) above.
♦ a will be present in the set but b will not be present.
7. Let us see an example:
• Consider the set: A = {x : x ∈ R, -3 ≤ x ≤ 5}
♦ This can be written in short form as: A = [-3, 5]
• Consider another set: B = {x : x ∈ R, -7 ≤ x ≤ 9}
♦ This can be written in short form as: B = [-7, 9]
• It is easy to visualize the number line containing the four points -3, 5, -7 and 9
♦ Obviously, the points -3 and 5 will lie within -7 and 9
• So all the infinite number of points from -3 to 5 will be present inside the set B
♦ Thus we get: A ⊂ B
8. Subsets of R:
• The infinite number of positive real numbers that we use in mathematics are members of the set: [0, ∞)
♦ This [0, ∞) is a subset of R
• The infinite number of negative real numbers that we use in mathematics are members of the set: (-∞, 0]
♦ This (-∞, 0] is a subset of R
• R can be represented as: (-∞, ∞)
9. The number (b - a) is called length of the interval [a, b]
• For the interval (a, b] also, the length is (b - a)
• This is because, even if we do not include a in the set, the length will begin from a point very close to a
• So in general, we can write:
(b - a) is the length of all the four intervals: [a, b], (a,b), (a, b] and [a, b)
Power Set
The power set can be explained using an example. It can be written in 9 steps:
1. Consider the set A = {1, 2}
• Let us write all the subsets of A.
2. We know that ɸ is a subset of every set.
• So ɸ is a subset of A.
3. {1} and {2} are subsets of A.
4. We know that, every set is a subset of itself.
• So in total, there are four subsets for A. They are:
ɸ, {1}, {2} and {1,2}
5. The set containing all those subsets is known as the power set of A.
We can write the definition:
| Definition 5: • The set containing all subsets of a set A is called power set of A. ♦ It is denoted as: p(A). |
6. In p(A), every element is a set.
• So in the present case, we can write: p(A) = {ɸ, {1}, {2}, {1,2}}
7. Recall that, number of elements in a set is denoted as n(A).
♦ So in our present case, we get n(A) = 2
8. In the same way, we will need to write the number of elements in p(A) also.
• The number of elements in a power set is denoted using square brackets.
• So in our present case, the number of elements in p(A) is denoted as n[p(A)].
♦ Obviously, n[p(A)] = 4
9. For any set A, the number n[p(A)] will be equal to 2m
♦ Where m = n(A)
◼ In other words,
♦ The number of elements in the power set of A is obtained by raising ‘2’ to a suitable number.
♦ This ‘suitable number’ is: The number of elements in A.
Universal Set
• This can be explained as follows:
• While doing problems in set theory, we will be dealing with:
♦ One basic set.
✰ Subsets of the basic set.
• For example, while dealing with simple numbers,
♦ The basic set may be Z.
✰ One subset may be even numbers.
✰ Another subset may be prime numbers.
• For more complex problems,
♦ The basic set may be R.
✰ One subset may be Q.
✰ Another subset may be T.
◼ Such a basic set (to which, all other sets are subsets) is called Universal set.
We will now see some solved examples. Link is given below:
In the next
section, we will see Venn diagrams
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