In everyday life, we often speak of collections of objects of a particular kind, such as, a pack of cards, a crowd of people, a cricket team, etc. In mathematics also, we come across collections, for example, of natural numbers, points, prime numbers, etc. More specially, we examine the following collections:
`(i)` Odd natural numbers less than `10,` i.e., `1, 3, 5, 7, 9`
`(ii)` The rivers of India
`(iii)` The vowels in the English alphabet, namely, `a, e, i, o, u`
`(iv)` Various kinds of triangles
`(v)` Prime factors of `210,` namely, `2,3,5` and `7`
`(vi)` The solution of the equation: `x^2 - 5x + 6 = 0`, viz, `2` and `3.`
We note that each of the above example is a well-defined collection of objects in the sense that we can definitely decide whether a given particular object belongs to a given collection or not. For example, we can say that the river Nile does not belong to the collection of rivers of India. On the other hand, the river Ganga does belong to this colleciton.
`text(We give below a few more examples of sets used particularly in mathematics, viz.)`
• N denotes set of all natural numbers= {1, 2, 3, ... }
• Z or I denotes set of all integers
= { ... '- 3,- 2, -1, 0, 1, 2, 3, ... }
• `Z_0` or `I_0` denotes set of all integers excluding zero
`= { ... ,- 3,- 2 ,'- 1, 1, 2, 3, ... }`
• `Z^(+)` or `I^(+)` denotes set of all positive integers
={1,2,3, ... }=N
• E denotes set of all even integers
= { ... ,- 6,- 4,- 2, 0, 2, 4, 6, ... }
• O denotes set of all odd integers
= { ... '- 5,- 3, -1, 1, 3, 5, ... }
• W denotes set of all whole numbers = {0, 1, 2, 3, ... }
• Q denotes set of all rational numbers= { x: x = pI q, where p and q are
integers and `q ne 0 }`.
• `Q_0` denotes set of all non-zero rational numbers `{x:x =p // q,` where p and q
are integers and `p ne 0` and `q ne 0 }.`
• `Q^(+)` denotes set of all positive rational numbers `= {x: x = p// q,` where p and q
are both positive or negative integers}
• R denotes set of all real numbers.
• `R_0` denotes set of all non-zero real numbers.
• `R^(+)` denotes set of all positive real numbers.
• R- Q denotes set of all irrational numbers.
• C denotes set of all complex numbers `= {a+ ib: a, b in R` and `i = sqrt(-1)}`
• `C_0` denotes set of all non··zero complex numbers
`= {a+ ib: a, b in R_0` and `i = sqrt(-1)}`
• `N_a` denotes set of all natural numbers which are less than or equal to a,
where a is positive integer
`text(The following points may be noted :)`
`(i)` Objects, elements and members of a set are synonymous terms.
`(ii)` Sets are usually denoted by capital letters `A, B, C, X, Y, Z,` etc.
`(iii)` The elements of a set are represented by small letters `a, b, c, x, y, z,` etc.
If `a` is an element of a set `A,` we say that - `a` belongs to `A`- the Greek symbol `in` is used to denote the phrase -belongs to-.Thus, we write `a in A`. If `-b-` is not an element of a set `A,` we write `b ∉ A` and read -`b` does not belong to `A-`.
In everyday life, we often speak of collections of objects of a particular kind, such as, a pack of cards, a crowd of people, a cricket team, etc. In mathematics also, we come across collections, for example, of natural numbers, points, prime numbers, etc. More specially, we examine the following collections:
`(i)` Odd natural numbers less than `10,` i.e., `1, 3, 5, 7, 9`
`(ii)` The rivers of India
`(iii)` The vowels in the English alphabet, namely, `a, e, i, o, u`
`(iv)` Various kinds of triangles
`(v)` Prime factors of `210,` namely, `2,3,5` and `7`
`(vi)` The solution of the equation: `x^2 - 5x + 6 = 0`, viz, `2` and `3.`
We note that each of the above example is a well-defined collection of objects in the sense that we can definitely decide whether a given particular object belongs to a given collection or not. For example, we can say that the river Nile does not belong to the collection of rivers of India. On the other hand, the river Ganga does belong to this colleciton.
`text(We give below a few more examples of sets used particularly in mathematics, viz.)`
• N denotes set of all natural numbers= {1, 2, 3, ... }
• Z or I denotes set of all integers
= { ... '- 3,- 2, -1, 0, 1, 2, 3, ... }
• `Z_0` or `I_0` denotes set of all integers excluding zero
`= { ... ,- 3,- 2 ,'- 1, 1, 2, 3, ... }`
• `Z^(+)` or `I^(+)` denotes set of all positive integers
={1,2,3, ... }=N
• E denotes set of all even integers
= { ... ,- 6,- 4,- 2, 0, 2, 4, 6, ... }
• O denotes set of all odd integers
= { ... '- 5,- 3, -1, 1, 3, 5, ... }
• W denotes set of all whole numbers = {0, 1, 2, 3, ... }
• Q denotes set of all rational numbers= { x: x = pI q, where p and q are
integers and `q ne 0 }`.
• `Q_0` denotes set of all non-zero rational numbers `{x:x =p // q,` where p and q
are integers and `p ne 0` and `q ne 0 }.`
• `Q^(+)` denotes set of all positive rational numbers `= {x: x = p// q,` where p and q
are both positive or negative integers}
• R denotes set of all real numbers.
• `R_0` denotes set of all non-zero real numbers.
• `R^(+)` denotes set of all positive real numbers.
• R- Q denotes set of all irrational numbers.
• C denotes set of all complex numbers `= {a+ ib: a, b in R` and `i = sqrt(-1)}`
• `C_0` denotes set of all non··zero complex numbers
`= {a+ ib: a, b in R_0` and `i = sqrt(-1)}`
• `N_a` denotes set of all natural numbers which are less than or equal to a,
where a is positive integer
`text(The following points may be noted :)`
`(i)` Objects, elements and members of a set are synonymous terms.
`(ii)` Sets are usually denoted by capital letters `A, B, C, X, Y, Z,` etc.
`(iii)` The elements of a set are represented by small letters `a, b, c, x, y, z,` etc.
If `a` is an element of a set `A,` we say that - `a` belongs to `A`- the Greek symbol `in` is used to denote the phrase -belongs to-.Thus, we write `a in A`. If `-b-` is not an element of a set `A,` we write `b ∉ A` and read -`b` does not belong to `A-`.