We can derive many relations from the given number of sets. Sets can be related with their own elements or with other sets. In this chapter, we shall discuss the various relations of a set with itself and with other sets. Such relations are unique.
Summary
Following are the various relations we can obtain between given sets:
- Subset
- Overlapping Sets
- Disjoint Sets
- Equal Sets
- Equivalent Sets
- Power Set
- Universal Set
Subset
The word ‘subset’ a word formed by adding a prefix ‘sub’ to the word ‘set’. We already know what set means. And, ‘sub’ means smaller units than the reference unit. So, a subset of a set A is another set, say set B, formed by the elements contained in set A. Mathematically, this relation is written as B ⊆ A or B ⊂ A, depending on the type of subset which we will discuss below.
Types of Subsets
There are two types of subsets depending on the comparison of their cardinality number with the parent set or super set.
- Proper Subset: A subset whose cardinal number is less than the cardinal number of its super set is called a proper subset. If set B is a proper subset of set A then, it is written as B ⊂ A.
- Improper Subset: A subset whose cardinal number is equal to the cardinal number of its parent set is called an improper subset. If set B is an improper subset of set A then, it is written as B ⊆ A.
Few Important Relations of Subsets
Here, n represents the cardinal number of the Super Set.
- Total number of subsets = 2n
- Total number of proper subsets = 2n-1
- Total number of improper subsets = 1
How to write subsets of a given set?
While writing subsets of a given set randomly might be very easy to you, listing down all the possible subsets might be a little tricky. So, follow the given steps to write them down properly:
- Identify: First, you need to make up your mind. You will be doing this by knowing how many subsets does the given set have. For this, you need to know the cardinality number of the given set then, apply the formula 2n. Now, you will need to find the given 2n number of subsets.
- Its easy: A common subset of every sets is a null set. Another common subset of sets other than a null set is the set itself. So, now you have two subsets of a given set.
- List down all the elements of the set individually.
- List down the possible combinations of the elements of the subsets. Start with two elements. Then, if all the possible subsets aren’t fulfilled, write down possible combinations of three elements. And so on.
At the end, you will have the desired number of subsets.
SOLVED EXAMPLE 1
Q. Find the subsets of the given set A = {1,2,3}.
Solution:
Given,
A = {1,2,3}
Step 1: Cardinal number of set A is 3, i.e. n = 3. So, 2n = 23 = 8. This means the given set A has 8 subsets.
Step 2: Two of the subsets are: {} and {1,2,3}. Now, we need to find 8-2 = 6 subsets only.
Step 3: Listing down the elements individually. We get: {1} and {2} and {3}
Step 4: Listing down the possible combinations of two elements together. We get {1,2} and {1,3} and {2,3}.
Step 5: Required subsets are: {}, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3} is the answer.
Overlapping Sets
Two or more sets are said to be overlapping if they have at least one element in common.
SOLVED EXAMPLE 2
Q. Are given sets A and B overlapping? A = {1,2,3} and B = {3,4,5}.
Solution:
Given,
A = {1,2,3}
B = {3,4,5}
We know,
For two sets to be overlapping, they must have at least one common element in them. Here, in the given sets A and B, element ‘3’ is common in both the sets. Hence, set A and set B are overlapping sets.
The other way of understanding overlapping sets is: They intersect each other. We will discuss more about intersection of sets below.
Disjoint Sets
Two ore more sets are said to be disjoint sets if they have no elements in common.
SOLVED EXAMPLE 3
Q. Are given sets A and B disjoin? A = {1,2,3} and B = {4,5,6}.
Solution:
Given,
A = {1,2,3}
B = {4,5,6}
We know,
For two sets to be disjoint, they must have no elements in common. Here, in the given sets A and B, each sets have distinct sets of elements that aren’t common with each other. Thus, we can say that the given sets are disjoint sets.
The other way of understanding overlapping sets is: They do not intersect each other. They are completely two spheres having their own individuality. Each of the sets have distinct elements with no alikeness in between them. But, they can have the same cardinal number.
GOOD TO KNOW
A disjoint set can never be overlapping set and vice-versa.
Equal Sets
Given sets are said to be equal sets when their cardinality number and elements are same.
When we take Set A as a reference set, for another set B or C to be considered as equal set with set A, should have exact same elements as contained in set A. They should not have either more or less number of elements. However, the order of appearance of elements in the sets can be different.
For example:
Set A = {a,b,c,d}
Set B = {b,c,d,a}
Set C = {c,d,a,b}
Set D = {d,c,a,b}
Set E = {d,c,b,a} and so on.
All above-mentioned sets are equal to each other because they contain exactly the same elements and same cardinal number (n = 4). But, you can see that the order of appearance of these elements are different. Even that, these are equal sets.
SOLVED EXAMPLE 4
Q. Are given sets A and B equal? A = {1,2,3} and B = {2,1,3}.
Solution:
Given,
A = {1,2,3}
B = {2,1,3}
We know,
For two sets to be equal, they must have same cardinal number and same elements. Since, these both conditions are satisfied by the sets given i.e. cardinal number of both sets A and B is n = 3 and they have the exact same elements i.e. {1, 2, 3}. Therefore, given sets A and B are equal.
We can conclude following things about equal sets:
- Considered sets must have same number of elements
- Considered sets must have the exact same elements. Order of appearance of the elements doesn’t make any difference.
Equivalent Sets
Given sets are said to be equivalent sets when they have the same cardinal number. For two sets to be equivalent, they should not have the same elements but they must have the same number of elements.
SOLVED EXAMPLE 5
Q. Are given sets A and B equivalent? A = {1,2,3,4,5} and B = {a,b,c,d,e}.
Solution:
Given,
A = {1,2,3,4,5}
B = {a,b,c,d,e}
Listing the cardinal numbers of given sets,
n(A) = 5
n(B) = 5
We know,
For two sets to be equivalent they must have the same cardinal number. Since sets A and B have the same cardinal numbers i.e. 5, they are equivalent sets.
An equivalent set has nothing to do with what is inside the sets. But, it has everything to do with how many members are there in the sets.
An equal set is an equivalent set but an equivalent set is not an equal set.
Power Set
We have discussed about subsets in the very beginning of this chapter. A power set is totally related to the concept of a subset. One must be able to list down the subsets of a given set in order to find a power set. Now, this leads us to a question: What is a power set? Well, a power set is a set whose elements are the subsets of a given set. In other words, a set containing all the possible subsets of a given set is called a power set.
With reference to SOLVED EXAMPLE 1, the power set of set A is P = { {}, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3} }.
Universal Set
Before getting into what is a universal set, can you remember what the word ‘universal’ mean? It is formed by the word ‘universe’ and we define universe as a vast collection of everything that is present in the space, right? Now, see there is a word set following the word universal! Can you guess what these two words mean when they are combined?
When we talk about more than one set, a new set containing all the elements contained in those reference sets is called a universal set. The other way of defining a universal set is: When a number of subsets can be produced from a single set, then that set is called the universal set. It means to say that all the elements that we are discussing must be contained within the universal set.
SOLVED EXAMPLE 6
Q. A = {1,2,3,4,5,b} and B = {a,b,c,d,e}, Find the universal set U.
Solution:
Given,
A = {1,2,3,4,5,b}
B = {a,b,c,d,e}
Universal set (U) = {1,2,3,4,5,a,b,c,d,e}
We know,
A universal set contains a number of elements, without repetition, that are present in its subsets also. This set might contain other elements than the elements mentioned in the given sets.
So, the answer of the above question can also be (U) = {1,2,3,4,5,6,7,8,a,b,c,d,e,f,g}.
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