We can use the following two methods to represent a set.
(i) Listing Method In this method, elements are listed and put within a braces { } and separated by commas.
This method is also known as Tabular method or Roster method.
e.g., A = Set of all prime numbers less than 11 = {2, 3, 5, 7}
(ii) Set Builder Method In this method, instead of listing all elements of a set, we list the property or properties satisfied by the elements of set and write it as
A = {x : P(x)} or {x | P(x)}
It is read as “A is the set of all elements x such that x has the property P x( ).” The symbol ‘:’ or ‘|’ stands for such that.
This method is also known as Rule method or Property method.
e.g., A = {1 ,2 ,3 ,4 ,5 ,6 ,7 ,8 } = {x : x $\in$ N and $\leq$ 8}
Note
- The order of elements in a set has no importance e.g., {1, 2, 3} and {3, 1, 2} are same sets.
- The repetition of elements in a set does not effect the set, e.g., {1, 2, 3} and {1, 1, 2, 3} both are same sets.
Notation of Some Standard Sets
(i) Set of all natural numbers, N = {1 ,2 ,3 , . . .}
(ii) Set of all whole numbers, W = {0 ,1 ,2 ,3 , . . .}
(iii) (a) Set of all integers, I or Z = {. . . , -2, -1 ,0 ,1 ,2 , . . .}
(b) Set of all positive or negative integers,
$I^+$ = {1 ,2 ,3 , . . . $\infty$} or $I^-$ = {-1 ,-2 ,-3 , . . . $\infty$}
(c) Set of all even (E) or odd (O) integers,
E = {. . . , -4, -2 ,0 ,2 ,4 , . . .} or O = {. . . , -3, -1 , 0, 1, 3, . . .}
(iv) (a) Set of all rational numbers,
Q = {p/q , where p and q are integers and q $\neq$ 0}
(b) Set of all irrational numbers, IR = {which cannot be p and I $\in$ I , q $\neq$ 0}
(c) Set of all real numbers,
R = {x: -$\infty$ < x < $\infty$}
(v) Set of all complex numbers,
C = {a + ib}, a, b $\in$ R and i = $\sqrt{-1}$
Sample Problem 1:
(1) Let P be the collection of all prime numbers. Then it can be represented in the set builder form as
P = {x |x is a prime number}
(2) Let X be the set of all even positive integers which are less than 15. Then
X = {x |x is even integer and 0 < x < 15}
X = {2, 4, 6, 8, 10, 12, 14}
(3) Let X be the set given above in (2) and
Y = {y|y = 0 or $\frac{1}{y}$ $\in$ X}
then
Y = {0, 1/2, 1/4, 1/6, 1/8, 1/10, 1/12, 1/14}
Sample Problem 2: [NCERT]
The builder form of following set is A = {3 ,6 ,9 ,12}, B = {1, 4, 9, 100..., }
(a) A = {x :x = 3n, n $\in$ N and 1 $\leq$ n $\leq$ 5},
B = {x :x = $n^2$, n $\in$ N and 1 $\leq$ n $\leq$ 10},
(b) A = {x :x = 3n, n $\in$ N and 1 $\leq$ n $\leq$ 4},
B = {x :x = $n^2$, n $\in$ N and 1 $\leq$ n $\leq$ 10},
(c) A = {x :x = 3n, n $\in$ N and 1 $\leq$ n $\leq$ 5},
B = {x :x = $n^2$, n $\in$ N and 1< n < 10},
(d) None of the above
Answer:
(b) Given A = {3 ,6 ,9 ,12} = {x :x = 3n, n $\in$ N and 1 $\leq$ n $\leq$ 4},
and B = {1, 4, 9, 100..., } = {x :x = $n^2$, n $\in$ N and 1 $\leq$ n $\leq$ 10}
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