Types of Sets
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A set is an unordered collection of objects, called elements or members of the set. Sets are the collection of objects whose elements are fixed and can not be changed. In other words, a set is well defined as the collection of data that does not carry from person to person. The elements can not be repeated in the set but can be written in any order. The set is represented by capital letters.
These objects are also known as elements of the set. The elements present in the set cannot be repeated in the set although can be written in any order. The set is denoted by capital letters.
Example: P = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
Components of a set are embedded in curly brackets distributed by commas as can be seen in the above example.
Types of Sets
Sets can be classified into many categories. Some of which are the finite set, infinite set, subset, universal set, proper set, power set, singleton set, empty set etc. Each type of set has its own importance during calculations. Basically, in our day-to-day life, sets are used to represent bulk data and collection of data. So, here in this article, we are going to learn and discuss the universal set.
1. Empty Sets or Null Sets
Empty set or Null set is a set that does not contain any element. The cardinality of the empty set is zero. The null set or the void set is expressed by the symbol ∅ and is read as phi. In roster form, ∅ is indicated by {}.
A common way of representing the null set is given by
∅ = { x : x ≠ x }, this set is empty, since there is no element which is not equal to itself. For example, a = a, 2 = 2.
An empty set Is said to be a finite set as the number of elements/symbols in an empty set is finite, i.e., zero(0).
Example:
S={x|x∈N and 7<x<8}=∅
2. Singleton Sets
A set consisting of only one element is sa Singleton Sets
A set consisting of only one element is said to be Singleton set.
Example:
• S= {x|x∈N, 7<x<9} = {8} is a Singleton set.
• Set S = {5} , M = {a} are said to be singleton since they are consists of only one element 5 and ‘a’ respectively. id to be Singleton set.
Example:
• S= {x|x∈N, 7<x<9} = {8} is a Singleton set.
• Set S = {5} , M = {a} are said to be singleton since they are consists of only one element 5 and ‘a’ respectively.
3. Finite Sets
A set that contains a finite number of elements is named a finite set. In other words, we can say that a set that includes no element or a definite number of elements is said to be a finite set. The empty set is also termed a finite set. Finite set contains exactly n distinct element where n is a non-negetive integer. Here, n is said to be “cardinality of sets.” The cardinality of sets is a natural number (∈ N) is said to be Finite set is denoted by|A|, # A, card (A) or n (A).
Example:
• Cardinality of empty set θ is 0 and is denoted by |θ| = 0
• Sets of even positive integer is not a finite set.
• The set of different colours in the rainbow is also an example of a finite set.
• Set P = {4,5,6,7,8,9,10} is a finite set, as it has a finite number of elements.
• M = {x : x ∈ M, x < 8}
• Q = {3, 5, 7, 11, 13, 17, 19 …… 113} are also examples of a finite set.
• Sets A = {a, b, c, d}, B = {5,7,9,15,78} and C = { x : x is a multiple of 3, where 0<x<100)
Here A, B and C all three contain a finite number of elements i.e. 4 in A, 5 in B and 33 in C and therefore will be called finite sets.
• A = { 5, 7, 9, 11} and B = { 4 , 8 , 16, 32, 64, 128}
Obviously, A, B contain a finite number of elements, i.e. 4 objects in A and 6 in B. Thus they are finite sets.
A set is called a finite set if there is one to one correspondence between the elements in the set and the element in some set n, where n is a natural number and n is the cardinality of the set. Finite Sets are also called numerable sets. N is termed as the cardinality of sets or a cardinal number of sets.
4. Infinite Sets
A set containing infinite number of elements i.e. whose cardinality can not found is said to be an Infinite set. Exactly opposite to the finite set, the infinite set will have an infinite number of elements. If a presented set is not finite, then it will be an infinite set.
OR
A set that has an infinite number of components is named an infinite set.
Thus, the set of all natural numbers. N = {1, 2, 3, 4 . . . .} is an infinite set.
Similarly,
- The set of all rational numbers between any two numbers will be infinite.
- B = {x : x ∈ B, x > 2}
- D = {x : x ∈ D, x = 3m}
- Set of all prime numbers, Set of all even numbers, Set of all odd numbers are examples of an infinite set.
- All infinite sets cannot be represented in roster form.
Example
• A = {x : x ∈ Q, 2 < x < 5} is an infinite set. Set A = {3,4,5,6,7} is a finite set, as it has a finite number of elements.
• Example – S={x|x∈N and x>10}
• Set R of all +ve real numbers less than 1 that can be represented by the decimal form 0. A1,a2,a3….. Where a1 is an integer such that 0 ≤ ai ≤ 9.
• Set C = {number of cows in India} is an infinite set, there is an approximate number of cows in India, but the actual number of cows cannot be expressed, as the numbers could be very large and counting all cows is not possible.
• The set of all rational number between ) and 1 given by
A = {x:x E Q, 0 <x<1} is an infinite set.
5. Subsets
Suppose A is a given set. Any set B, each of whose elements is also an element of A, is called contained in A and is said to be a subset of A.
The symbol ⊆ stands for “is contained in” or “is subset of”. Thus, if “B is contained in A” or “B is subset of A”, we write
B ⊆ A.
When B is subset of A, we also say ‘A contains B’ or ‘A is superset of B.
The symbol ⊇ is read for “contains” this A ⊇ B means “A contains B”.
- The symbol ‘⊆’ is applied to signify ‘is a subset of’ or ‘is included in’.
- A ⊆ B; implies A is a subset of B in other words A is contained in B.
- B ⊆ A; implies B is a subset A.
- Every given set is a subset of itself.
Example:
• If A= {1, 2} and B= {4, 2, 1} the A is the subset of B or A ⊆ B.
• Set A= {p, q, r, s, t, u}
Set B= {m, n, o, p, q, r, s, t, u}
Then we can state A ⊆ B.
Let us take another example; X = { 3, 4, 5, 6, 7, 8, 9 } and Y = { 6, 7 }. Hereabouts we can see that set Y is a subset of set X as all the components of set Y are in set X. Therefore, we can write Y ⊆ X.
• If A = (3, 5, 7), B = (3, 5, 7, 9) than A ⊆ B since every element of A is also an element of B. But B ⊄ A since 9 ∈ B while 9 ∉ A.
Properties of Subsets:
- Every set is a subset of itself.
- The Null Set i.e.∅ is a subset of every set.
- If A is a subset of B and B is a subset of C, then A will be the subset of C. If A⊂B and B⊂ C ⟹ A ⊂ C
- A finite set having n elements has 2n subsets.
6. Proper Subset
If A is a subset of B and A ≠ B then A is said to be a proper subset of B. If A is a proper subset of B then B is not a subset of A, i.e., there is at least one element in B which is not in A. In other words, if each element of B is an element of A and there is at least one element of A which is not an element of B, then B is said to be a proper subsets of A. “Is proper subset of” is symbolically represented by ⊂.
Also, the empty set ∅ is a proper subset of every set except itself.
Example:
• Let A = {2, 3, 4}
B = {2, 3, 4, 5}
A is a proper subset of B.
• If A = {2, 3, 4, 7, 8}
Here n(A) = 5
B = {1, 2, 3, 4, 7, 8, 10}
Here n(B) = 7
We can observe that all the elements of A are present in B but the element ‘1, 10’ of B is not available in A. Therefore, we say that A is a proper subset of B. Symbolically, this is written as A ⊂ B.
7. Improper Subset
If A is a subset of B and A = B, then A is said to be an improper subset of B.
Example:
• A = {2, 3, 4}, B = {2, 3, 4}
A is an improper subset of B.
• Every set is an improper subset of itself.
8. Power Sets
The power of any given set A is the set of all subsets of A and is denoted by P (A). If A has n elements, then P (A) has 2n elements. The power of any given set A is the set of all subsets of A and is denoted by P (A). The number of components of the power set is given by.
That is for a set A which covers n elements, the total number of subsets that can be created is . From this, we can state that P(A) will have elements.
Example:
• {} is a subset of {2,3}
{2} is a subset of {2,3}
{3} is a subset of {2,3}
{2,3} is also a subset of {2,3}
Therefore, power set of X = {2,3},P(X) = {{},{2},{3},{2,3}}
• The set A= {a, b, c} then its subsets are ∅, {a}, {b}, {c}, {a, b}, {a, c}, {b, c} {a, b, c}.
Therefore, P(A) = {∅, {a}, {b}, {c}, {a, b}, {a, c}, {b, c} {a, b, c} }.
9. Equal Sets
When two sets consists of same elements, whether in the same order, they are said to be equal. In other words any two sets are declared to be equal sets if and only if they are equivalent and as well as their elements are identical. Two sets A and B are said to be equal and written as A = B if both have the same elements. Therefore, every element which belongs to A is also an element of the set B and every element which belongs to the set B is also an element of the set A.
A = B ⟺ {x ϵ A ⟺ x ϵ B}.
If there is some element in set A that does not belong to set B or vice versa then A ≠ B, i.e., A is not equal to B.
OR
Two sets P and Q are supposed to be equal if they hold the same elements. Each element of P is an element of Q and every element of Q is an element of P.
Example:
• Let P{1, 2, 3, 4, 5} and Q={y : y , for 0<y<6 , y ∈ natural numbers}
Writing Q in the tabular form {1, 2, 3, 4, 5}
Here, each element of P is an element of Q, i.e., P ⊆ Q.
Also, every element of Q is an element of P, i.e., Q ⊆ P.
Therefore, sets P and Q stand for equal sets.
• Let A = {3,4,5,6} and B = {6,5,4,3}, then A = Band if A = {set of even numbers} and B = { set of natural numbers} then A ≠ B, because natural numbers consist of all the positive integers starting from 1, 2, 3, 4, 5 to infinity, but even numbers start with 2, 4, 6, 8, and so on.
• A = {1,2,3,4,5} and B = {1,5,2,4,3} , then A = B.
• Let A = {3,4,5,6} and B = {6,5,4,3}, then A = Band if A = {set of even numbers} and B = { set of natural numbers} then A ≠ B, because natural numbers consist of all the positive integers starting from 1, 2, 3, 4, 5 to infinity, but even numbers start with 2, 4, 6, 8, and so on.
10. Equivalent Sets
If the cardinalities of two sets are equal, they are called equivalent sets. In other words, the number of different elements in a given set A is termed as the cardinal number of A and is denoted by n(A).
If A {y : y ∈ N, x < 7}
The set A = {1, 2, 3, 4, 5, 6}
Therefore, n(A) = 6
Similarly;
P = set of letters in the word TESTBOOK.
P = {T, E, S, T, B, O, O, K}
Therefore, n(P) = 8.
Any two sets are stated to be equivalent sets if their cardinality is the same.
OR
Two sets P and Q are supposed to be equivalent if their cardinal number is identical, i.e., n(P) = n(Q). The symbol for expressing an equivalent set is ‘↔’.
Example:
• A = {3, 2, 5} Here n(A) = 3.
B = {r, s, t} Here n(B) = 3.
Therefore, A ↔ B.
• If A= {1, 2, 6} and B= {16, 17, 22}, they are equivalent as cardinality of A is equal to the cardinality of B. i.e. |A|=|B|=3
11. Universal Sets
A set which consists of all the elements of the considering sets is said to be the Universal set for those sets.
It is generally denoted by U or S.
This is the set that is the foundation for every other set developed. Depending upon the circumstances, the universal set is chosen. It may be a finite or infinite set. All the other sets remain the subsets of the universal set.
Example:
• For the set of all integers, the Universal Set can be the set of rational numbers.
• In the human population studies the universal set consists of all the people in the world.
• Consider if set A = {2,3,4}, set B = {4,5,6,7} and C = {6,7,8,9, 10}
Then, we will address the universal set as U = {2,3,4,5,6,7,8,9,10}
• Set A = {1,2,3}, set B = {3,4,5,6}, and C = {5,6,7,8,9}.Then, we will write the universal set as, U = {1,2,3,4,5,6,7,8,9,}.
• Consider the following sets, A = {a, b, c, d, e} ; B = {x, y, z} and U = {a, b, c, d, e, f, g, h, w, x, y, z} and B, since U contains all the elements of A and B.
• Here, U is the universal set for A Let A = {1, 2, 3}
C = { 0, 1} then we can take
S = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} as universal set.
Note: As per the definition of the universal set, we can say that all the sets are subsets of the universal set.
Hence,
A ⊆ U
B ⊆ U
And C ⊆ U.
12. Disjoint Sets
Two sets A and B are called disjoint sets if they do not have even one element in common. Therefore, Two sets A and B are said to be disjoint if no element of A is in B and no element of B is in A.
Therefore, disjoint sets have the following properties –
- N(A∩B)=∅
- N(A∪B)=n(A)+n(B)
- If two sets X and Y do not have any common elements, and their intersection results in zero(0), then set X and Y are called disjoint sets. It can be represented as;, X ∩ Y = 0.
Example:
• R = {a, b, c}
S = {k, p, m}
R and S are disjoint sets.
• Let, A={1,2,6} and B={7,9,14}, there is not a single common element, hence these sets are disjoint sets.
Sets Symbols
Set symbols are used to define the elements of a given set. The table presents some of these symbols with their meaning.
Partitions of a Set
Let S be a nonempty set. A partition of S is a subdivision of S into nonoverlapping, nonempty subsets. Speceficially, a partition of S is a collection {Ai} of nonempty subsets of S such that:
Aj≠ Ak Then Aj ∩ Ak= ∅
The subsets in a partition are called cells.
Fig: Venn diagram of a partition of the rectangular set S of points into five cells,A1,A2,A3,A4,A5
Venn Diagrams
Venn diagram, invented in 1880 by John Venn, is a schematic diagram that shows all possible logical relations between different mathematical sets. Venn diagram is a pictorial representation of sets in which an enclosed area in the plane represents sets.
Examples:
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