2.SETS
In mathematics set is defined as the collection of well defined object which can be separated distinctly.
For instance,
S = {2, 4, 6, 8} is collection of the even integers.
A set can be explained in different ways:
i. Listing method: A = {a, b, c, .z}
ii. Descriptive method: N = {the natural numbersfrom 1 to 50}
iii. Set builder method: A – B= A -(A∩ B )
iv. Venn – diagram
Universal sets
A universal set is the collection of all objects in a particular context or theory. All other sets in that framework constitute subsets of the universal set, which is denoted as letter U. The objects themselves are known as elements or members of U.
Subsets
The set made by elements of the universal sets is called subsets of the universal sets
For example
U = {1, 2, 3, 4, …………..50}
A = {even integers from 1 to 50}
B= {odd numbers from 1 to 50}
Here, A and B are the subsets of U
Overlapping sets
Two sets are said to be overlapped if they have same element in common.
A∩ B = {6}
A and B are overlapping sets.
Disjoint sets
Two sets are said to be disjoint sets if there is no element in common.
Cardinality of the sets
The number of the elements in the given sets is known as cardinality of sets.
A = {1, 2, 5,}
B = {5, 3, 4}
AUB = {1, 2, 3, 4, 5}
n(A) =3
n(B) = 3
n(AUB) = 5
Cardinality of the three sets
LetA and B and Crepresent threesets asa shown in the figure s
n(AUBUC) =n(A) + n(B) + n(C) - n(A∩ B) -- n(B∩ C) - n(C∩ A) +n(A∩ B∩ C)
Operation of sets
Union of sets
The set which includes elements of A and B is called union of the sets.
A = {1, 2, 5,}
B = {5, 3, 4}
AUB = {1, 2, 3, 4, 5}
Or, AUB= {x: xϵ A orxϵ B}
n(A) =3
n(B) = 3
n(AUB) = n(A) + n(B) - n(A∩ B)
= 3 +3 -1 = 5
Intersections of sets
If the elements of set belongs to both Sets A and B, it is called intersection of A and B.
A = {1, 2, 5}
B = {5, 3, 4}
A∩ B = {5}
n(A∩ B) = n(A) – n0 (A) = 3 -2 = 1
Or, A∩ B = {x: xϵ A and xϵ B}
Complement of sets
The set that contains all the elements of universal sets except the given set A is called complement of the set A . It is denoted by A̅
Difference of sets
If A and B are the two sets , the difference of the dsets is the elements of the set thst includes only in one set .
A – B= A -(A∩ B )
B- A= B-(A∩ B )
Example
Examples1
In a group of 200 students who like game, 120 like cricket game an 105 like football game. By drawing Venn diagram find
i. how many students like both the games ?
ii. How many students like only cricket?
Soln
n(U) = 200
C and F denote the students who study Cricket and football respectively.
n(C) =120
n(F) = 105
n( C ∩ F)=?
We have
n(CUF)=n(C) + n(F) -n(C∩ F)
200=120 + 105 -n(C∩ F)
n(C∩ F)= 25
n0(C) = n(C)-n( C ∩ F)= 120 – 25 = 95
Examples 2
In the certain examination, 50% students passed in account, 30% passed in English, 30% failed in both and 25 student passes in both subjects. By drawing Venn – diagram, find the number of the students who passes in account only.
Let total number be x A and E denotes the students who study account and English respectively
n(A) =50%
n(E) = 30%
n( A U E)c = 30%
n( A U E) = 25
We have
n(U)= n(A)+ n(E)+ - n( A ∩E)
100%= 50 % + 30% + 30 % - n( A ∩E)
n( A ∩E) = 10%
According to the question,
10% of x = 25
x= 250
The number of the students who passed in accounts only = 40% of the 250 = 100