`(1)` `text(Null Set or Empty Set or Void Set)`
A set having no element is called a null set or empty set or void set. It is
denoted by `phi` or { }.
`text(Note)`
1. `phi` is called the null set.
2. `phi` is unique.
3. `phi` is a subset of every set.
4. `phi` is never written within braces. i.e., {} is not the null set.
5. {0} is not an empty set as it contains the element 0 (zero).
For example,
`1. {x : x in N, 4 < x< 5} =phi`
2. {x:xE R,x2 + 1 = 0} = ¢
`3. {x : x^2 = 25, x` is even number} `= phi`
`(2)` `text(Singleton or Unit Set)`
A set having one and only one element is called singleton or unit set.
For example, `{x: x- 3 = 4}` is a singleton set.
Since, `x - 3 = 4`
`x= 7`
` therefore {x: x- 3 = 4} = {7}`
`(3)` `text(Subset)`
If every element of a set A is also an element of a set B, then A is called the
subset of B, we write `A ⊑ B` (read as A is subset of B or A is contained in B).
Thus, `A ⊑ B , [x in A => x in B]`
`text(Note)`
1. Every set is a subset of itself
i.e., `A ⊑ A`
2. If `A ⊑ B, B ⊑ C,` then `A ⊑ C.`
For example,
1. If A= {2, 3, 4} and B= {b, 4, 2, 3, 1}, then `A ⊑ B`.
2. The sets {a}, { b }, {a, b }, { b, c} are the subsets of the set {a, b, c }.
`(4)text( Finite and Infinite Sets)`
Let `A = {1, 2, 3, 4, 5},\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ B = {a, b, c, d, e, g}`
and` C = { text(men living presently in different parts of the world)}`
We observe that `A` contains `5` elements and `B` contains `6` elements. How many elements does `C` contain `?` As it is, we do not know the number of elements in `C,` but it is some natural number which may be quite a big number. By number of elements of a set `S,` we mean the number of distinct elements of the set and we denote it by `n (S).` If `n (S)` is a natural number, then `S` is non-empty finite set.
Consider the set of natural numbers. We see that the number of elements of this set is not finite since there are infinite number of natural numbers. We say that the set of natural numbers is an infinite set. The sets `A`, `B` and `C` given above are finite sets and `n(A)= 5, n(B) = 6` and `n(C) = text(some finite number.)`
`text(Note):`All infinite sets cannot be described in the roster form. For example, the set of real numbers cannot be described in this form, because the elements of this set do not follow any particular pattern.
`(5)text( Equal Sets :)`
Given two sets `A` and `B,` if every element of `A` is also an element of `B` and if every element of `B` is also an element of `A,` then the sets `A` and `B` are said to be equal. Clearly, the two sets have exactly the same elements.
`text(Note):``A` set does not change if one or more elements of the set are repeated. For example, the sets `A = {1, 2, 3}` and `B = {2, 2, 1, 3, 3}` are equal, since each element of `A` is in `B` and vice-versa. That is why we generally do not repeat any element in describing a set.
`(6)text(Power Set)`
Consider the set `{1, 2}.` Let us write down all the subsets of the set `{1, 2}.` We know that `phi` is a subset of every set . So, `phi` is a subset of `{1, 2}.` We see that `{1}` and `{ 2 }` are also subsets of `{1, 2}.` Also, we know that every set is a subset of itself. So, `{ 1, 2 }` is a subset of `{1, 2}.` Thus, the set `{ 1, 2 }` has, in all, four subsets, viz. `phi, { 1 }, { 2 } and { 1, 2 }.` The set of all these subsets is called the power set of `{ 1, 2 }.`
`(7)` `text(Super Set)`
The statement `A ⊑ B` can be rewritten as `B ⊒ A,` then B is called the super set of A and is written as `B supset A.`
`(8)text(Universal Set):`
Usually, in a particular context, we have to deal with the elements and subsets of a basic set which is relevant to that particular context. For example, while studying the system of numbers, we are interested in the set of natural numbers and its subsets such as the set of all prime numbers, the set of all even numbers, and so forth. This basic set is called the `text(Universal Set.)` The universal set is usually denoted by `U,` and all its subsets by the letters `A, B, C,` etc.
`(1)` `text(Null Set or Empty Set or Void Set)`
A set having no element is called a null set or empty set or void set. It is
denoted by `phi` or { }.
`text(Note)`
1. `phi` is called the null set.
2. `phi` is unique.
3. `phi` is a subset of every set.
4. `phi` is never written within braces. i.e., {} is not the null set.
5. {0} is not an empty set as it contains the element 0 (zero).
For example,
`1. {x : x in N, 4 < x< 5} =phi`
2. {x:xE R,x2 + 1 = 0} = ¢
`3. {x : x^2 = 25, x` is even number} `= phi`
`(2)` `text(Singleton or Unit Set)`
A set having one and only one element is called singleton or unit set.
For example, `{x: x- 3 = 4}` is a singleton set.
Since, `x - 3 = 4`
`x= 7`
` therefore {x: x- 3 = 4} = {7}`
`(3)` `text(Subset)`
If every element of a set A is also an element of a set B, then A is called the
subset of B, we write `A ⊑ B` (read as A is subset of B or A is contained in B).
Thus, `A ⊑ B , [x in A => x in B]`
`text(Note)`
1. Every set is a subset of itself
i.e., `A ⊑ A`
2. If `A ⊑ B, B ⊑ C,` then `A ⊑ C.`
For example,
1. If A= {2, 3, 4} and B= {b, 4, 2, 3, 1}, then `A ⊑ B`.
2. The sets {a}, { b }, {a, b }, { b, c} are the subsets of the set {a, b, c }.
`(4)text( Finite and Infinite Sets)`
Let `A = {1, 2, 3, 4, 5},\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ B = {a, b, c, d, e, g}`
and` C = { text(men living presently in different parts of the world)}`
We observe that `A` contains `5` elements and `B` contains `6` elements. How many elements does `C` contain `?` As it is, we do not know the number of elements in `C,` but it is some natural number which may be quite a big number. By number of elements of a set `S,` we mean the number of distinct elements of the set and we denote it by `n (S).` If `n (S)` is a natural number, then `S` is non-empty finite set.
Consider the set of natural numbers. We see that the number of elements of this set is not finite since there are infinite number of natural numbers. We say that the set of natural numbers is an infinite set. The sets `A`, `B` and `C` given above are finite sets and `n(A)= 5, n(B) = 6` and `n(C) = text(some finite number.)`
`text(Note):`All infinite sets cannot be described in the roster form. For example, the set of real numbers cannot be described in this form, because the elements of this set do not follow any particular pattern.
`(5)text( Equal Sets :)`
Given two sets `A` and `B,` if every element of `A` is also an element of `B` and if every element of `B` is also an element of `A,` then the sets `A` and `B` are said to be equal. Clearly, the two sets have exactly the same elements.
`text(Note):``A` set does not change if one or more elements of the set are repeated. For example, the sets `A = {1, 2, 3}` and `B = {2, 2, 1, 3, 3}` are equal, since each element of `A` is in `B` and vice-versa. That is why we generally do not repeat any element in describing a set.
`(6)text(Power Set)`
Consider the set `{1, 2}.` Let us write down all the subsets of the set `{1, 2}.` We know that `phi` is a subset of every set . So, `phi` is a subset of `{1, 2}.` We see that `{1}` and `{ 2 }` are also subsets of `{1, 2}.` Also, we know that every set is a subset of itself. So, `{ 1, 2 }` is a subset of `{1, 2}.` Thus, the set `{ 1, 2 }` has, in all, four subsets, viz. `phi, { 1 }, { 2 } and { 1, 2 }.` The set of all these subsets is called the power set of `{ 1, 2 }.`
`(7)` `text(Super Set)`
The statement `A ⊑ B` can be rewritten as `B ⊒ A,` then B is called the super set of A and is written as `B supset A.`
`(8)text(Universal Set):`
Usually, in a particular context, we have to deal with the elements and subsets of a basic set which is relevant to that particular context. For example, while studying the system of numbers, we are interested in the set of natural numbers and its subsets such as the set of all prime numbers, the set of all even numbers, and so forth. This basic set is called the `text(Universal Set.)` The universal set is usually denoted by `U,` and all its subsets by the letters `A, B, C,` etc.