10th Class SSC Mathematics: Sets

1. In Mathematics, Set theory was developed by George Cantor (1845 – 1918).

2. Set: A well defined collections of objects is called a Set.

3. Well defined means that:
(i) All the objects in the Set should have a common feature or property and
(ii) It should be possible to decide whether any given objects belongs to the set or not.

4. We usually denote a set by capital letters and the elements of a set are represented
by small letters.

5. Ex: Set of vowels in English language V = { a, e, i, o, u }
i. Set of even numbers E = { 2, 4, 6, 8, ……….}
ii. Set of odd numbers O = {1, 3, 5, 7, …………}
iii. Set of prime numbers P = {2, 3, 5, 7, 11, 13, …..}

6. Any element or object belonging to a set, then we use symbol ‘∈’ (belongs to), if it
is not belonging to it is denoted by the Symbol ‘∉’(does not belongs to)
Ex: In Natural numbers Set N, 1∈ N and 0 ∉ N.

7. Roaster Form: All elements are written in order by separating commas and are
enclosed with in curly brackets is called Roaster form. In the form elements should
not repeated.

8. Ex: Set of prime numbers less than 13 is p ={ 2, 3,5, 7, 11 }.

9. Set Builder Form: In set builder form, we use a symbol x (or any other symbol y, z etc.) for the element of the set. This is following by a colon (or a vertical line), after which we write the characteristic property possessed within curly brackets.

10. Ex: P = { 2, 3, 5, 7, 11}. This is the set of all prime numbers less than 13. It can be represented in the set builders form as P = { x: x is a prime numbers less than 13 } (Or) P= { x/x is a prime number less than 13}.

11. Null Set: A set which does not contain any element is called the empty set or the null get or a void set .It is denoted by φ or { } Ex: A = { x/ 1-2 =0 and x is a rational number}.

12. Finite Set: A set is called a finite set if it is possible to count the numbers of elements in it. Ex: A = { x ; x∈ N and (x-1)(x-2) =0} = {1,2} B = {x; x is a day in a week} = {SUN, MON, TUS, WED, THU, FRI, SAT}.

13. Infinite Set: A Set is called an infinite set if the number of cannot count the number of elements in it. Ex: A = {x / x∈ N and x is an odd number} = { 1, 3, 5, 7, 9 ,11 ………..} B = { x/ x is a point on a straight line}.

14. Cardinal Number: The number of elements in a Set is called the cardinal number of the set. If ‘A’ is a set them n(A) represents cardinal number. Ex: If A= {a, e, i, o, u } then n(A) =5If B = {x; x is alter in the word INDIA}. Then n(B) = 4 N (φ)= 0.

15. Universal Set: Universal Set is denoted by ‘µ’ or ‘U’ generally , universal set represented by rectangle.

16. Subset: If every element of a set A is also an element of set B, then the set A is said to be a subset of set B. It is represented as A ⊂ B. Ex: If A ={4, 8,12 }; B = {2, 4, 6, 8, 10, 12, 14} then A is a subset of B (i.e. A⊂ B).

i. Every Set is a subset of itself (A⊂ Α).

ii. Empty Set is a subset of every set (φ ⊂ Α).

iii. If A ⊂ B and B⊂ C then A⊂ C (Transitive property).

17. Equal Sets: Two sets A and B are said to be ‘equal’ if every elements in A belongs to B and every elements in B belongs to A. If A and B are equal sets, then we write A=B. Ex: The set of prime number less than 6, A= {2, 3, 5}.The prime factors of 30, B = {2, 3, 5}.